Three-Dimensional Transformations

3D transformation methods are extended from 2D methods by including considerations for the z coordinate

A 3D homogenous coordinate is represented as a four-element column vector Each geometric transformation operator is a 4 by 4 matrix

Geometric Transformations in Three-Dimensional Space

3D Translation Translation of a Point

zyx tzztyytxx

11000100010001

1

zyx

ttt

zyx

z

y

x

Prsquo=PT

(xyz) T=(txtytz)

(xrsquoyrsquozrsquo)

x

z

y

3D Scaling Uniform Scaling (Scaling relative to the coordinate

origin)

zyx szzsyysxx

11000000000000

1

zyx

ss

s

zyx

z

y

x

xz

y Prsquo = SP

Relative Scaling Scaling with a Selected fixed point (xf yf zf)

11000100010001

1000000000000

1000100010001

1

)()()(zyx

zyx

ss

s

zyx

zyx

zyxTsssSzyxTf

f

f

z

y

x

f

f

f

fffzyxfff

x x x xzzzz

y y y y

Original position Translate Scaling Inverse Translate

3D Rotation Positive rotation angles produce counterclockwise rotations

about a coordinate axis assuming that we are looking in the negative direction along that coordinate axis

Coordinate-Axis Rotations Z-axis rotation X-axis rotation Y-axis rotation

General 3D Rotations Rotation about an axis that is parallel to one of the coordinate

axes Rotation about an arbitrary axis

Coordinate-Axis Rotations Z-Axis Rotation X-Axis Rotation Y-Axis Rotation

11000010000cossin00sincos

1

zyx

zyx

110000cossin00sincos00001

1

zyx

zyx

110000cos0sin00100sin0cos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

or Prsquo = Rz(θ)P

11000010000cossin00sincos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

Other axis rotations xrarr yrarr zrarr x

x-axis rotation yrsquo=y cos θ - z sin θ zrsquo=y sin θ + z cos θ xrsquo=x

y-axis rotation zrsquo=z cos θ - x sin θ xrsquo=z sin θ + x cos θ yrsquo=y

or Prsquo = Rx(θ)P

or Prsquo = Ry(θ)P

General 3D Rotations CASE 1bull Rotation about an Axis that is Parallel to One of the Coordinate Axes

ndash Translate the object so that the rotation axis coincides with the parallel coordinate axis

ndash Perform the specified rotation about that axisndash Translate the object so that the rotation axis is moved back to its original position

ndash Any coordinate position P on the object in this fig is transformed with the sequence shown as below

Prsquo = T-1Rx(θ)TP

General 3D Rotations CASE 2 Rotation about an Arbitrary Axis

Basic Idea1 Translate (x1 y1 z1) to

the origin2 Rotate (xrsquo2 yrsquo2 zrsquo2) on to

the z axis3 Rotate the object around

the z-axis4 Rotate the axis to the

original orientation5 Translate the rotation axis

to the original position

(x2y2z2)

(x1y1z1)

x

z

y

R-1

T-1

R

T

TRRRRRTR xyzyx111

General 3D Rotations Step 1 Translation

1000100010001

1

1

1

zyx

T

(x2y2z2)

(x1y1z1)

x

z

y

General 3D Rotations Step 2 Establish [ TR ]

x x axis

100000000001

10000cossin00sincos00001

dcdbdbdc

x

R

(abc)(0bc)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z

cgvrkoreaackr

Arbitrary Axis Rotation Step 3 Rotate about y axis by

(abc)

(a0d)

ld

22

222222

cossin

cbd

dacballd

la

100000001000

10000cos0sin00100sin0cos

ldla

lald

y

Rx

y

Projected Point

zRotated Point

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

3D transformation methods are extended from 2D methods by including considerations for the z coordinate

A 3D homogenous coordinate is represented as a four-element column vector Each geometric transformation operator is a 4 by 4 matrix

Geometric Transformations in Three-Dimensional Space

3D Translation Translation of a Point

zyx tzztyytxx

11000100010001

1

zyx

ttt

zyx

z

y

x

Prsquo=PT

(xyz) T=(txtytz)

(xrsquoyrsquozrsquo)

x

z

y

3D Scaling Uniform Scaling (Scaling relative to the coordinate

origin)

zyx szzsyysxx

11000000000000

1

zyx

ss

s

zyx

z

y

x

xz

y Prsquo = SP

Relative Scaling Scaling with a Selected fixed point (xf yf zf)

11000100010001

1000000000000

1000100010001

1

)()()(zyx

zyx

ss

s

zyx

zyx

zyxTsssSzyxTf

f

f

z

y

x

f

f

f

fffzyxfff

x x x xzzzz

y y y y

Original position Translate Scaling Inverse Translate

3D Rotation Positive rotation angles produce counterclockwise rotations

about a coordinate axis assuming that we are looking in the negative direction along that coordinate axis

Coordinate-Axis Rotations Z-axis rotation X-axis rotation Y-axis rotation

General 3D Rotations Rotation about an axis that is parallel to one of the coordinate

axes Rotation about an arbitrary axis

Coordinate-Axis Rotations Z-Axis Rotation X-Axis Rotation Y-Axis Rotation

11000010000cossin00sincos

1

zyx

zyx

110000cossin00sincos00001

1

zyx

zyx

110000cos0sin00100sin0cos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

or Prsquo = Rz(θ)P

11000010000cossin00sincos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

Other axis rotations xrarr yrarr zrarr x

x-axis rotation yrsquo=y cos θ - z sin θ zrsquo=y sin θ + z cos θ xrsquo=x

y-axis rotation zrsquo=z cos θ - x sin θ xrsquo=z sin θ + x cos θ yrsquo=y

or Prsquo = Rx(θ)P

or Prsquo = Ry(θ)P

General 3D Rotations CASE 1bull Rotation about an Axis that is Parallel to One of the Coordinate Axes

ndash Translate the object so that the rotation axis coincides with the parallel coordinate axis

ndash Perform the specified rotation about that axisndash Translate the object so that the rotation axis is moved back to its original position

ndash Any coordinate position P on the object in this fig is transformed with the sequence shown as below

Prsquo = T-1Rx(θ)TP

General 3D Rotations CASE 2 Rotation about an Arbitrary Axis

Basic Idea1 Translate (x1 y1 z1) to

the origin2 Rotate (xrsquo2 yrsquo2 zrsquo2) on to

the z axis3 Rotate the object around

the z-axis4 Rotate the axis to the

original orientation5 Translate the rotation axis

to the original position

(x2y2z2)

(x1y1z1)

x

z

y

R-1

T-1

R

T

TRRRRRTR xyzyx111

General 3D Rotations Step 1 Translation

1000100010001

1

1

1

zyx

T

(x2y2z2)

(x1y1z1)

x

z

y

General 3D Rotations Step 2 Establish [ TR ]

x x axis

100000000001

10000cossin00sincos00001

dcdbdbdc

x

R

(abc)(0bc)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z

cgvrkoreaackr

Arbitrary Axis Rotation Step 3 Rotate about y axis by

(abc)

(a0d)

ld

22

222222

cossin

cbd

dacballd

la

100000001000

10000cos0sin00100sin0cos

ldla

lald

y

Rx

y

Projected Point

zRotated Point

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

3D Translation Translation of a Point

zyx tzztyytxx

11000100010001

1

zyx

ttt

zyx

z

y

x

Prsquo=PT

(xyz) T=(txtytz)

(xrsquoyrsquozrsquo)

x

z

y

3D Scaling Uniform Scaling (Scaling relative to the coordinate

origin)

zyx szzsyysxx

11000000000000

1

zyx

ss

s

zyx

z

y

x

xz

y Prsquo = SP

Relative Scaling Scaling with a Selected fixed point (xf yf zf)

11000100010001

1000000000000

1000100010001

1

)()()(zyx

zyx

ss

s

zyx

zyx

zyxTsssSzyxTf

f

f

z

y

x

f

f

f

fffzyxfff

x x x xzzzz

y y y y

Original position Translate Scaling Inverse Translate

3D Rotation Positive rotation angles produce counterclockwise rotations

about a coordinate axis assuming that we are looking in the negative direction along that coordinate axis

Coordinate-Axis Rotations Z-axis rotation X-axis rotation Y-axis rotation

General 3D Rotations Rotation about an axis that is parallel to one of the coordinate

axes Rotation about an arbitrary axis

Coordinate-Axis Rotations Z-Axis Rotation X-Axis Rotation Y-Axis Rotation

11000010000cossin00sincos

1

zyx

zyx

110000cossin00sincos00001

1

zyx

zyx

110000cos0sin00100sin0cos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

or Prsquo = Rz(θ)P

11000010000cossin00sincos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

Other axis rotations xrarr yrarr zrarr x

x-axis rotation yrsquo=y cos θ - z sin θ zrsquo=y sin θ + z cos θ xrsquo=x

y-axis rotation zrsquo=z cos θ - x sin θ xrsquo=z sin θ + x cos θ yrsquo=y

or Prsquo = Rx(θ)P

or Prsquo = Ry(θ)P

General 3D Rotations CASE 1bull Rotation about an Axis that is Parallel to One of the Coordinate Axes

ndash Translate the object so that the rotation axis coincides with the parallel coordinate axis

ndash Perform the specified rotation about that axisndash Translate the object so that the rotation axis is moved back to its original position

ndash Any coordinate position P on the object in this fig is transformed with the sequence shown as below

Prsquo = T-1Rx(θ)TP

General 3D Rotations CASE 2 Rotation about an Arbitrary Axis

Basic Idea1 Translate (x1 y1 z1) to

the origin2 Rotate (xrsquo2 yrsquo2 zrsquo2) on to

the z axis3 Rotate the object around

the z-axis4 Rotate the axis to the

original orientation5 Translate the rotation axis

to the original position

(x2y2z2)

(x1y1z1)

x

z

y

R-1

T-1

R

T

TRRRRRTR xyzyx111

General 3D Rotations Step 1 Translation

1000100010001

1

1

1

zyx

T

(x2y2z2)

(x1y1z1)

x

z

y

General 3D Rotations Step 2 Establish [ TR ]

x x axis

100000000001

10000cossin00sincos00001

dcdbdbdc

x

R

(abc)(0bc)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z

cgvrkoreaackr

Arbitrary Axis Rotation Step 3 Rotate about y axis by

(abc)

(a0d)

ld

22

222222

cossin

cbd

dacballd

la

100000001000

10000cos0sin00100sin0cos

ldla

lald

y

Rx

y

Projected Point

zRotated Point

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

3D Scaling Uniform Scaling (Scaling relative to the coordinate

origin)

zyx szzsyysxx

11000000000000

1

zyx

ss

s

zyx

z

y

x

xz

y Prsquo = SP

Relative Scaling Scaling with a Selected fixed point (xf yf zf)

11000100010001

1000000000000

1000100010001

1

)()()(zyx

zyx

ss

s

zyx

zyx

zyxTsssSzyxTf

f

f

z

y

x

f

f

f

fffzyxfff

x x x xzzzz

y y y y

Original position Translate Scaling Inverse Translate

3D Rotation Positive rotation angles produce counterclockwise rotations

about a coordinate axis assuming that we are looking in the negative direction along that coordinate axis

Coordinate-Axis Rotations Z-axis rotation X-axis rotation Y-axis rotation

General 3D Rotations Rotation about an axis that is parallel to one of the coordinate

axes Rotation about an arbitrary axis

Coordinate-Axis Rotations Z-Axis Rotation X-Axis Rotation Y-Axis Rotation

11000010000cossin00sincos

1

zyx

zyx

110000cossin00sincos00001

1

zyx

zyx

110000cos0sin00100sin0cos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

or Prsquo = Rz(θ)P

11000010000cossin00sincos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

Other axis rotations xrarr yrarr zrarr x

x-axis rotation yrsquo=y cos θ - z sin θ zrsquo=y sin θ + z cos θ xrsquo=x

y-axis rotation zrsquo=z cos θ - x sin θ xrsquo=z sin θ + x cos θ yrsquo=y

or Prsquo = Rx(θ)P

or Prsquo = Ry(θ)P

General 3D Rotations CASE 1bull Rotation about an Axis that is Parallel to One of the Coordinate Axes

ndash Translate the object so that the rotation axis coincides with the parallel coordinate axis

ndash Perform the specified rotation about that axisndash Translate the object so that the rotation axis is moved back to its original position

ndash Any coordinate position P on the object in this fig is transformed with the sequence shown as below

Prsquo = T-1Rx(θ)TP

General 3D Rotations CASE 2 Rotation about an Arbitrary Axis

Basic Idea1 Translate (x1 y1 z1) to

the origin2 Rotate (xrsquo2 yrsquo2 zrsquo2) on to

the z axis3 Rotate the object around

the z-axis4 Rotate the axis to the

original orientation5 Translate the rotation axis

to the original position

(x2y2z2)

(x1y1z1)

x

z

y

R-1

T-1

R

T

TRRRRRTR xyzyx111

General 3D Rotations Step 1 Translation

1000100010001

1

1

1

zyx

T

(x2y2z2)

(x1y1z1)

x

z

y

General 3D Rotations Step 2 Establish [ TR ]

x x axis

100000000001

10000cossin00sincos00001

dcdbdbdc

x

R

(abc)(0bc)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z

cgvrkoreaackr

Arbitrary Axis Rotation Step 3 Rotate about y axis by

(abc)

(a0d)

ld

22

222222

cossin

cbd

dacballd

la

100000001000

10000cos0sin00100sin0cos

ldla

lald

y

Rx

y

Projected Point

zRotated Point

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Relative Scaling Scaling with a Selected fixed point (xf yf zf)

11000100010001

1000000000000

1000100010001

1

)()()(zyx

zyx

ss

s

zyx

zyx

zyxTsssSzyxTf

f

f

z

y

x

f

f

f

fffzyxfff

x x x xzzzz

y y y y

Original position Translate Scaling Inverse Translate

3D Rotation Positive rotation angles produce counterclockwise rotations

about a coordinate axis assuming that we are looking in the negative direction along that coordinate axis

Coordinate-Axis Rotations Z-axis rotation X-axis rotation Y-axis rotation

General 3D Rotations Rotation about an axis that is parallel to one of the coordinate

axes Rotation about an arbitrary axis

Coordinate-Axis Rotations Z-Axis Rotation X-Axis Rotation Y-Axis Rotation

11000010000cossin00sincos

1

zyx

zyx

110000cossin00sincos00001

1

zyx

zyx

110000cos0sin00100sin0cos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

or Prsquo = Rz(θ)P

11000010000cossin00sincos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

Other axis rotations xrarr yrarr zrarr x

x-axis rotation yrsquo=y cos θ - z sin θ zrsquo=y sin θ + z cos θ xrsquo=x

y-axis rotation zrsquo=z cos θ - x sin θ xrsquo=z sin θ + x cos θ yrsquo=y

or Prsquo = Rx(θ)P

or Prsquo = Ry(θ)P

General 3D Rotations CASE 1bull Rotation about an Axis that is Parallel to One of the Coordinate Axes

ndash Translate the object so that the rotation axis coincides with the parallel coordinate axis

ndash Perform the specified rotation about that axisndash Translate the object so that the rotation axis is moved back to its original position

ndash Any coordinate position P on the object in this fig is transformed with the sequence shown as below

Prsquo = T-1Rx(θ)TP

General 3D Rotations CASE 2 Rotation about an Arbitrary Axis

Basic Idea1 Translate (x1 y1 z1) to

the origin2 Rotate (xrsquo2 yrsquo2 zrsquo2) on to

the z axis3 Rotate the object around

the z-axis4 Rotate the axis to the

original orientation5 Translate the rotation axis

to the original position

(x2y2z2)

(x1y1z1)

x

z

y

R-1

T-1

R

T

TRRRRRTR xyzyx111

General 3D Rotations Step 1 Translation

1000100010001

1

1

1

zyx

T

(x2y2z2)

(x1y1z1)

x

z

y

General 3D Rotations Step 2 Establish [ TR ]

x x axis

100000000001

10000cossin00sincos00001

dcdbdbdc

x

R

(abc)(0bc)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z

cgvrkoreaackr

Arbitrary Axis Rotation Step 3 Rotate about y axis by

(abc)

(a0d)

ld

22

222222

cossin

cbd

dacballd

la

100000001000

10000cos0sin00100sin0cos

ldla

lald

y

Rx

y

Projected Point

zRotated Point

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

3D Rotation Positive rotation angles produce counterclockwise rotations

about a coordinate axis assuming that we are looking in the negative direction along that coordinate axis

Coordinate-Axis Rotations Z-axis rotation X-axis rotation Y-axis rotation

General 3D Rotations Rotation about an axis that is parallel to one of the coordinate

axes Rotation about an arbitrary axis

Coordinate-Axis Rotations Z-Axis Rotation X-Axis Rotation Y-Axis Rotation

11000010000cossin00sincos

1

zyx

zyx

110000cossin00sincos00001

1

zyx

zyx

110000cos0sin00100sin0cos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

or Prsquo = Rz(θ)P

11000010000cossin00sincos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

Other axis rotations xrarr yrarr zrarr x

x-axis rotation yrsquo=y cos θ - z sin θ zrsquo=y sin θ + z cos θ xrsquo=x

y-axis rotation zrsquo=z cos θ - x sin θ xrsquo=z sin θ + x cos θ yrsquo=y

or Prsquo = Rx(θ)P

or Prsquo = Ry(θ)P

General 3D Rotations CASE 1bull Rotation about an Axis that is Parallel to One of the Coordinate Axes

ndash Translate the object so that the rotation axis coincides with the parallel coordinate axis

ndash Perform the specified rotation about that axisndash Translate the object so that the rotation axis is moved back to its original position

ndash Any coordinate position P on the object in this fig is transformed with the sequence shown as below

Prsquo = T-1Rx(θ)TP

General 3D Rotations CASE 2 Rotation about an Arbitrary Axis

Basic Idea1 Translate (x1 y1 z1) to

the origin2 Rotate (xrsquo2 yrsquo2 zrsquo2) on to

the z axis3 Rotate the object around

the z-axis4 Rotate the axis to the

original orientation5 Translate the rotation axis

to the original position

(x2y2z2)

(x1y1z1)

x

z

y

R-1

T-1

R

T

TRRRRRTR xyzyx111

General 3D Rotations Step 1 Translation

1000100010001

1

1

1

zyx

T

(x2y2z2)

(x1y1z1)

x

z

y

General 3D Rotations Step 2 Establish [ TR ]

x x axis

100000000001

10000cossin00sincos00001

dcdbdbdc

x

R

(abc)(0bc)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z

cgvrkoreaackr

Arbitrary Axis Rotation Step 3 Rotate about y axis by

(abc)

(a0d)

ld

22

222222

cossin

cbd

dacballd

la

100000001000

10000cos0sin00100sin0cos

ldla

lald

y

Rx

y

Projected Point

zRotated Point

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Coordinate-Axis Rotations Z-Axis Rotation X-Axis Rotation Y-Axis Rotation

11000010000cossin00sincos

1

zyx

zyx

110000cossin00sincos00001

1

zyx

zyx

110000cos0sin00100sin0cos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

or Prsquo = Rz(θ)P

11000010000cossin00sincos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

Other axis rotations xrarr yrarr zrarr x

x-axis rotation yrsquo=y cos θ - z sin θ zrsquo=y sin θ + z cos θ xrsquo=x

y-axis rotation zrsquo=z cos θ - x sin θ xrsquo=z sin θ + x cos θ yrsquo=y

or Prsquo = Rx(θ)P

or Prsquo = Ry(θ)P

General 3D Rotations CASE 1bull Rotation about an Axis that is Parallel to One of the Coordinate Axes

ndash Translate the object so that the rotation axis coincides with the parallel coordinate axis

ndash Perform the specified rotation about that axisndash Translate the object so that the rotation axis is moved back to its original position

ndash Any coordinate position P on the object in this fig is transformed with the sequence shown as below

Prsquo = T-1Rx(θ)TP

General 3D Rotations CASE 2 Rotation about an Arbitrary Axis

Basic Idea1 Translate (x1 y1 z1) to

the origin2 Rotate (xrsquo2 yrsquo2 zrsquo2) on to

the z axis3 Rotate the object around

the z-axis4 Rotate the axis to the

original orientation5 Translate the rotation axis

to the original position

(x2y2z2)

(x1y1z1)

x

z

y

R-1

T-1

R

T

TRRRRRTR xyzyx111

General 3D Rotations Step 1 Translation

1000100010001

1

1

1

zyx

T

(x2y2z2)

(x1y1z1)

x

z

y

General 3D Rotations Step 2 Establish [ TR ]

x x axis

100000000001

10000cossin00sincos00001

dcdbdbdc

x

R

(abc)(0bc)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z

cgvrkoreaackr

Arbitrary Axis Rotation Step 3 Rotate about y axis by

(abc)

(a0d)

ld

22

222222

cossin

cbd

dacballd

la

100000001000

10000cos0sin00100sin0cos

ldla

lald

y

Rx

y

Projected Point

zRotated Point

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

or Prsquo = Rz(θ)P

11000010000cossin00sincos

1

zyx

zyx

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

Other axis rotations xrarr yrarr zrarr x

x-axis rotation yrsquo=y cos θ - z sin θ zrsquo=y sin θ + z cos θ xrsquo=x

y-axis rotation zrsquo=z cos θ - x sin θ xrsquo=z sin θ + x cos θ yrsquo=y

or Prsquo = Rx(θ)P

or Prsquo = Ry(θ)P

General 3D Rotations CASE 1bull Rotation about an Axis that is Parallel to One of the Coordinate Axes

ndash Translate the object so that the rotation axis coincides with the parallel coordinate axis

ndash Perform the specified rotation about that axisndash Translate the object so that the rotation axis is moved back to its original position

ndash Any coordinate position P on the object in this fig is transformed with the sequence shown as below

Prsquo = T-1Rx(θ)TP

General 3D Rotations CASE 2 Rotation about an Arbitrary Axis

Basic Idea1 Translate (x1 y1 z1) to

the origin2 Rotate (xrsquo2 yrsquo2 zrsquo2) on to

the z axis3 Rotate the object around

the z-axis4 Rotate the axis to the

original orientation5 Translate the rotation axis

to the original position

(x2y2z2)

(x1y1z1)

x

z

y

R-1

T-1

R

T

TRRRRRTR xyzyx111

General 3D Rotations Step 1 Translation

1000100010001

1

1

1

zyx

T

(x2y2z2)

(x1y1z1)

x

z

y

General 3D Rotations Step 2 Establish [ TR ]

x x axis

100000000001

10000cossin00sincos00001

dcdbdbdc

x

R

(abc)(0bc)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z

cgvrkoreaackr

Arbitrary Axis Rotation Step 3 Rotate about y axis by

(abc)

(a0d)

ld

22

222222

cossin

cbd

dacballd

la

100000001000

10000cos0sin00100sin0cos

ldla

lald

y

Rx

y

Projected Point

zRotated Point

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

3D Rotation z-axis rotation

xrsquo=x cos θ - y sin θ yrsquo=x sin θ + y cos θ zrsquo=z

Other axis rotations xrarr yrarr zrarr x

x-axis rotation yrsquo=y cos θ - z sin θ zrsquo=y sin θ + z cos θ xrsquo=x

y-axis rotation zrsquo=z cos θ - x sin θ xrsquo=z sin θ + x cos θ yrsquo=y

or Prsquo = Rx(θ)P

or Prsquo = Ry(θ)P

General 3D Rotations CASE 1bull Rotation about an Axis that is Parallel to One of the Coordinate Axes

ndash Translate the object so that the rotation axis coincides with the parallel coordinate axis

ndash Perform the specified rotation about that axisndash Translate the object so that the rotation axis is moved back to its original position

ndash Any coordinate position P on the object in this fig is transformed with the sequence shown as below

Prsquo = T-1Rx(θ)TP

General 3D Rotations CASE 2 Rotation about an Arbitrary Axis

Basic Idea1 Translate (x1 y1 z1) to

the origin2 Rotate (xrsquo2 yrsquo2 zrsquo2) on to

the z axis3 Rotate the object around

the z-axis4 Rotate the axis to the

original orientation5 Translate the rotation axis

to the original position

(x2y2z2)

(x1y1z1)

x

z

y

R-1

T-1

R

T

TRRRRRTR xyzyx111

General 3D Rotations Step 1 Translation

1000100010001

1

1

1

zyx

T

(x2y2z2)

(x1y1z1)

x

z

y

General 3D Rotations Step 2 Establish [ TR ]

x x axis

100000000001

10000cossin00sincos00001

dcdbdbdc

x

R

(abc)(0bc)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z

cgvrkoreaackr

Arbitrary Axis Rotation Step 3 Rotate about y axis by

(abc)

(a0d)

ld

22

222222

cossin

cbd

dacballd

la

100000001000

10000cos0sin00100sin0cos

ldla

lald

y

Rx

y

Projected Point

zRotated Point

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

General 3D Rotations CASE 1bull Rotation about an Axis that is Parallel to One of the Coordinate Axes

ndash Translate the object so that the rotation axis coincides with the parallel coordinate axis

ndash Perform the specified rotation about that axisndash Translate the object so that the rotation axis is moved back to its original position

ndash Any coordinate position P on the object in this fig is transformed with the sequence shown as below

Prsquo = T-1Rx(θ)TP

General 3D Rotations CASE 2 Rotation about an Arbitrary Axis

Basic Idea1 Translate (x1 y1 z1) to

the origin2 Rotate (xrsquo2 yrsquo2 zrsquo2) on to

the z axis3 Rotate the object around

the z-axis4 Rotate the axis to the

original orientation5 Translate the rotation axis

to the original position

(x2y2z2)

(x1y1z1)

x

z

y

R-1

T-1

R

T

TRRRRRTR xyzyx111

General 3D Rotations Step 1 Translation

1000100010001

1

1

1

zyx

T

(x2y2z2)

(x1y1z1)

x

z

y

General 3D Rotations Step 2 Establish [ TR ]

x x axis

100000000001

10000cossin00sincos00001

dcdbdbdc

x

R

(abc)(0bc)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z

cgvrkoreaackr

Arbitrary Axis Rotation Step 3 Rotate about y axis by

(abc)

(a0d)

ld

22

222222

cossin

cbd

dacballd

la

100000001000

10000cos0sin00100sin0cos

ldla

lald

y

Rx

y

Projected Point

zRotated Point

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

General 3D Rotations CASE 2 Rotation about an Arbitrary Axis

Basic Idea1 Translate (x1 y1 z1) to

the origin2 Rotate (xrsquo2 yrsquo2 zrsquo2) on to

the z axis3 Rotate the object around

the z-axis4 Rotate the axis to the

original orientation5 Translate the rotation axis

to the original position

(x2y2z2)

(x1y1z1)

x

z

y

R-1

T-1

R

T

TRRRRRTR xyzyx111

General 3D Rotations Step 1 Translation

1000100010001

1

1

1

zyx

T

(x2y2z2)

(x1y1z1)

x

z

y

General 3D Rotations Step 2 Establish [ TR ]

x x axis

100000000001

10000cossin00sincos00001

dcdbdbdc

x

R

(abc)(0bc)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z

cgvrkoreaackr

Arbitrary Axis Rotation Step 3 Rotate about y axis by

(abc)

(a0d)

ld

22

222222

cossin

cbd

dacballd

la

100000001000

10000cos0sin00100sin0cos

ldla

lald

y

Rx

y

Projected Point

zRotated Point

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

General 3D Rotations Step 1 Translation

1000100010001

1

1

1

zyx

T

(x2y2z2)

(x1y1z1)

x

z

y

General 3D Rotations Step 2 Establish [ TR ]

x x axis

100000000001

10000cossin00sincos00001

dcdbdbdc

x

R

(abc)(0bc)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z

cgvrkoreaackr

Arbitrary Axis Rotation Step 3 Rotate about y axis by

(abc)

(a0d)

ld

22

222222

cossin

cbd

dacballd

la

100000001000

10000cos0sin00100sin0cos

ldla

lald

y

Rx

y

Projected Point

zRotated Point

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

General 3D Rotations Step 2 Establish [ TR ]

x x axis

100000000001

10000cossin00sincos00001

dcdbdbdc

x

R

(abc)(0bc)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z

cgvrkoreaackr

Arbitrary Axis Rotation Step 3 Rotate about y axis by

(abc)

(a0d)

ld

22

222222

cossin

cbd

dacballd

la

100000001000

10000cos0sin00100sin0cos

ldla

lald

y

Rx

y

Projected Point

zRotated Point

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

cgvrkoreaackr

Arbitrary Axis Rotation Step 3 Rotate about y axis by

(abc)

(a0d)

ld

22

222222

cossin

cbd

dacballd

la

100000001000

10000cos0sin00100sin0cos

ldla

lald

y

Rx

y

Projected Point

zRotated Point

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Arbitrary Axis Rotation Step 4 Rotate about z axis by the desired

angle

l

1000010000cossin00sincos

zR

y

x

z

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Arbitrary Axis Rotation Step 5 Apply the reverse transformation to

place the axis back in its initial position

x

y

l

l

z

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Find the new coordinates of a unit cube 90ordm-rotated about an axis defined by its endpoints A(210) and B(331)

A Unit Cube

Example

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Example Step1 Translate point A to the origin

Arsquo(000)x

z

y

Brsquo(121)

1000010010102001

T

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lBrsquo(121)

Projected point (021)

Brdquo(105)

Example Step 2 Rotate axis ArsquoBrsquo about the x axis by

and angle until it lies on the xz plane

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

Brdquo(10 5)(006)

Example Step 3 Rotate axis ArsquoBrsquorsquo about the y axis by

and angle until it coincides with the z axis

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Example Step 4 Rotate the cube 90deg about the z axis

Finally the concatenated rotation matrix about the arbitrary axis AB becomes

TRRRRRTR xyzyx 90111

1000010000010010

90zR

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

100056001670741065001511075066707420

7421983007501660

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

PRP

111111110760091056007260817065003011467148304090151122511840258048405580

89129091742172528162834166716502

11111111100110010000111111001100

100056001670741065001511075066707420

7421983007501660

P

Example Multiplying R(θ) by the point matrix of the original

cube

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

24

A 3-D Reflection can be performed relative to a selected reflection axis or wrt selected reflection plane The 3-D reflection matrixes are set up similarly to those for 2-D

In 2-D Reflection wrt axis is equivalent to 180 degree rotations about the axis in 3- D space

whereas in 3-D Reflection wrt a plane are equivalent to 180 degree rotations in 4-D space

3D Transformation

Other Transformations REFLECTION

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Other Transformations REFLECTION Reflection Relative to the XY Plane

xz

y

x

z

y

11000010000100001

1

zyx

zyx

Reflection Relative to the XZ Plane

11000010000100001

1

zyx

zyx

xz

yx

zy

11000010000100001

1

zyx

zyx

Reflection Relative to the YZ Plane

xz

y

z

y

x

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Other Transformations SHEARINGbull Shearing transformation are used to modify the shape of the

object and they are useful in 3-D viewing for obtaining General Projection transformations

bull Z-axis 3-D Shear transformation

bull The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value

while leaving z co-ordinate unchanged Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value

110000100010001

1

zyx

ba

zyx

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Other Transformations SHEARING

X-axis 3-D Shear transformation

Y-axis 3-D Shear transformation

110000100010001

1

zyx

b

a

zyx

110000100010001

1

zyx

ba

zyx

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

3D Projection

3D Transformation Slide 28

Viewing in 3D

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Principle Axisbull Man-made objects often have ldquocube-likerdquo shape

These objects have 3 principle axis

3D Transformation Slide 29

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

3D Transformation Slide 30

Projections

bull How do we map 3D objects to 2D spaceDisplay device (a screen) is 2Dhellip

bull 2D window to world and a viewport on the 2D surface

bull Clip what wont be shown in the 2D window and map the remainder to the viewport

2D to 2D is straight

forwardhellip

bull Solution Transform 3D objects on to a 2D plane using projections

3D to 2D is more complicatedhellip

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Projections

bull In 3Dhellipndash View volume in the worldndash Projection onto the 2D projection planendash A viewport to the view surface

bull Processhellipndash 1hellip clip against the view volume ndash 2hellip project to 2D plane or windowndash 3hellip map to viewport

3D Transformation Slide 31

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

32

Projections

bull Conceptual Model of the 3D viewing process

3D Transformation

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

33

PROJECTIONS

PARALLEL

(parallel projectors)PERSPECTIVE

(converging projectors)

One point(one principal vanishing point)

Two point(Two principal vanishing point)

Three point(Three principal vanishing point)

Orthographic(projectors perpendicular to view plane)

Oblique(projectors not perpendicular to view plane)

General

Cavalier

Cabinet

Multiview(view plane parallel to principal planes)

Axonometric(view plane not parallel to principal planes)

Isometric Dimetric Trimetric

3D Transformation

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Types of projectionsbull 2 types of projections

ndash PERSPECTIVE and PARALLEL

bull Key factor is the center of projection ndash if distance to center of projection is finite PERSPECTIVEndash if distance to center of projection is infinite PARALLEL

3D Transformation Slide 34

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

35

In perspective projection object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection)

In parallel projection coordinate positions are transformed to the view plane along parallel lines

3D Transformation

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

bull Perspective projection+ Size varies inversely with distance - looks realisticndash Distance and angles are not (in general) preservedndash Parallel lines do not (in general) remain parallel

bull Parallel projection+ Good for exact measurements+ Parallel lines remain parallelndash Angles are not (in general) preservedndash Less realistic looking

Perspective Vs Parallel

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Road in perspective

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

38

Perspective Projections

CHARACTERISTICS

bull Center of Projection (CP) is a finite distance from objectbull Projectors are rays (ie non-parallel)bull Vanishing pointsbull Objects appear smaller as distance from CP (eye of observer)

increasesbull Difficult to determine exact size and shape of objectbull Most realistic difficult to execute

3D Transformation

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

39

bull When a 3D object is projected onto view plane using perspective transformation equations any set of parallel lines in the object that are not parallel to the projection plane converge at a vanishing point ndash There are an infinite number of vanishing points

depending on how many set of parallel lines there are in the scene

bull If a set of lines are parallel to one of the three principle axes the vanishing point is called an principle vanishing point ndash There are at most 3 such points corresponding to the

number of axes cut by the projection plane

3D Transformation

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

40

bull Certain set of parallel lines appear to meet at a different pointndash The Vanishing point for this direction

bull Principle vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x y z axes

bull The number of principal vanishing points is determined by the number of principal axes intersected by the view plane

bull Sets of parallel lines on the same plane lead to collinear vanishing points ndash The line is called the horizon for that plane

Vanishing points

3D Transformation

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

41

Classes of Perspective Projection

bull One-Point Perspectivebull Two-Point Perspectivebull Three-Point Perspective

3D Transformation

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

42

One-Point Perspective

3D Transformation

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

43

Two-point perspective projection

3D Transformation

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

44

Three-point perspective projection

bull Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

3D Transformation

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Affine Transformationsbull Affine transformations are combinations of hellip

ndash Linear transformations andndash Translations

bull Properties of affine transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines remain parallelndash Ratios are preservedndash Closed under composition

wyx

fedcba

wyx

100

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Perspective Transformationsbull Projective transformations hellip

ndash Affine transformations andndash Projective warps

bull Properties of projective transformationsndash Origin does not necessarily map to originndash Lines map to linesndash Parallel lines do not necessarily remain parallelndash Ratios are not preservedndash Closed under composition

wyx

ihgfedcba

wyx

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

473D Transformation

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

483D Transformation

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

493D Transformation

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

503D Transformation

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

513D Transformation

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Center of projection is at infinity Direction of projection (DOP) same for all points

Parallel Projection

DOP

ViewPlane

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

53

bull We can define a parallel projection with a projection vector that defines the direction for the projection lines

2 types bull Orthographic when the projection is perpendicular to the view

plane In short ndash direction of projection = normal to the projection planendash the projection is perpendicular to the view plane

bull Oblique when the projection is not perpendicular to the view plane In short ndash direction of projection normal to the projection planendash Not perpendicular

Parallel Projections

3D Transformation

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

54

when the projection is perpendicular to the view plane

when the projection is not perpendicular to the view plane

bull Orthographic projection Oblique projection

3D Transformation

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

55

ndash Front side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view

ndash all have projection plane perpendicular to a principle axes

ndash Here length and angles are accurately depicted and measured from the drawing so engineering and architectural drawings commonly employee this

bull However As only one face of an object is shown it can be hard to create a mental image of the object even when several views are available

Orthographic (or orthogonal) projections

3D Transformation

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

56

Orthogonal projections

3D Transformation

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

57

Axonometric orthographic projections

The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance

3D Transformation

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

58

OBLIQUE PARALLEL PROJECTIONS

3D Transformation

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

59

Cavalier projectionbull All lines perpendicular to the projection plane are

projected with no change in length

OBLIQUE PARALLEL PROJECTIONS Cavalier and Cabinet

3D Transformation

bull The direction of the projection makes a 45 degree angle with the projection plane

bull Because there is no foreshortening this causes an exaggeration of the z axes

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Oblique Projections CAVALIER PROJECTIONbull DOP not perpendicular to view plane

Cavalier(DOP = 45o)tan() = 1

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

61

Cabinet projectionndash Lines which are perpendicular to the projection plane

(viewing surface) are projected at 1 2 the length ndash This results in foreshortening of the z axis and

provides a more ldquorealisticrdquo viewndash The direction of the projection makes a 634 degree

angle with the projection plane This results in foreshortening of the z axis and provides a more ldquorealisticrdquo view

3D Transformation

Oblique Projections CABINET PROJECTION

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

Oblique Projections CABINET PROJECTION

HampB

bull DOP not perpendicular to view plane

Cabinet(DOP = 634o)tan() = 2

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63

633D Transformation

Remaining Topics - (REFER CLASS NOTES)

Transformation Matrix for Oblique Projection of a 3-D point

General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation

View Volumes for Projections

• Three-Dimensional Transformations
• Geometric Transformations in Three-Dimensional Space
• 3D Translation
• 3D Scaling
• Relative Scaling
• 3D Rotation
• Coordinate-Axis Rotations
• 3D Rotation (2)
• 3D Rotation (3)
• General 3D Rotations CASE 1
• General 3D Rotations CASE 2
• General 3D Rotations
• General 3D Rotations (2)
• Arbitrary Axis Rotation
• Arbitrary Axis Rotation (2)
• Arbitrary Axis Rotation (3)
• Example
• Example (2)
• Example (3)
• Example (4)
• Example (5)
• Example (6)
• Example (7)
• Other Transformations REFLECTION
• Other Transformations REFLECTION (2)
• Other Transformations SHEARING
• Other Transformations SHEARING (2)
• 3D Projection
• Principle Axis
• Projections
• Projections
• Projections (2)
• Slide 33
• Types of projections
• In perspective projection object position are transforme
• Perspective Vs Parallel
• Road in perspective
• Perspective Projections
• Slide 39
• Vanishing points
• Classes of Perspective Projection
• One-Point Perspective
• Two-point perspective projection
• Three-point perspective projection
• Affine Transformations
• Perspective Transformations
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Parallel Projection
• Parallel Projections
• Slide 54
• Orthographic (or orthogonal) projections
• Orthogonal projections
• Axonometric orthographic projections
• Slide 58
• Slide 59
• Oblique Projections CAVALIER PROJECTION
• Slide 61
• Oblique Projections CABINET PROJECTION
• Slide 63