Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous...

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Two-Dimensional Geometric Transformations ch5. 참참 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations Properties of Transformation matrix

Transcript of Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous...

Page 1: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Two-Dimensional Geometric Transformations ch5. 참조

Subjects :

Basic Transformations

Homogeneous Coordinates

Composite Transformations

Other Transformations

Properties of Transformation matrix

Page 2: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Basic Transformations : Translation (1/2)

Translation (이동 ) Definition : repositioning objects along a straight line path

from one position to another Let : original position, : new position

We need translation distance (translation vector)

for x direction

for y direction Then,

y

x=P

y

x = P

T =t

t

x

y

x' x t

y' y t

x

y

Page 3: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Translation (2/2)

In matrix form

Translation is rigid-body Transformation that moves  objects without deformation

• every point on the object is translated by the same amount For straight line

• applying translation distance to each line end points For polygon, curves

P' P Tx

y

t

t

x

y

Page 4: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Basic Transformations : Rotation (1/3)

Rotation Definition : repositioning objects along a circular path in the xy plan

e

We need to specify• rotation angle

• rotation point (pivot point)

• direction (clockwise (-), counter clockwise(+))

a rotation about a rotation axis Z

Page 5: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Rotation (2/3)

x' rcos( ) rcos cos rsin sin

y' rsin( rcos sin rsin cos

)

r x y2 2

x rcos ,

x' xcos ysin

y' xsin ycos

y rsin

• Rotation angle : • rotation point : origin (0,0)• direction : c.c.w

Then

• where • original points are • so,

Page 6: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Rotation (3/3)

In matrix form

called Rotation matrix

Rotation about an arbitrary pivot position

As with translations, rotations are

rigid-body transformations

y

x

cossin

sincosPR P

-

==

x' x (x x )cos (y y )sin

y' y (x x )sin (y y )cos

r r r

r r r

Page 7: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Basic Transformations : scaling (1/2)

Scaling Definition : alters the size of an objects

we need scaling factors

: for x value

: for y value

In matrix form

Then

if : Uniform scaling

  : differential scaling

SS 0

0 S

X

y

x x sx' y y sy'

y

x

s0

0sPS P

y

x

s s

s s

x y

x y

Page 8: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

scaling (2/2)

if : uniform compression• move objects to the coordinate origin

if : uniform Enlargement• move objects farther from the origin

Fixed point scaling : scaling based on a fixed point• An object is scaled relative to the fixed point by scaling distance

from each vertex to fixed point

• when

0 s s 1x y

1 s sx y

( ),x yf f

x' x (x x )s x s x (1 s )

y' y (y y )s y s y (1 s )

f f x x f x

f f y y f y

(x ,y ) (0,0)f f

Page 9: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Homogeneous Coordinates (1/4)

Let’s consider combination of two transformation, translation after rotation

We can combine two matrices into a single matrix

by expanding the 2x2 matrix to 3x3 matrix

i.e.

21 MPMP

y

x

t

t

y

x

cossin

sincos

x y t

x y t

x

y

cos sin

sin cos

cos sin

sin cos

? ? ?

t

t

x

y

Page 10: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Homogeneous Coordinates (2/4)

To utilize above 3x3 matrix,• represent Cartesian coordinate point (x,y) with the Homogeneous

Coordinate (xh, yh, h)

• where

Homogeneous CoordinateHomogeneous Coordinate : (xh, yh, h) • first developed in geometry (1946)

• applied in graphics by Roberts (1965)

• usually h=1 (called weight value)

• then (2,3) = (2,3,1) = (1,1.5,0.5)

xx

h,y

y

h

h h

Page 11: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Homogeneous Coordinates (3/4)

• there is infinite number of equivalent representation for point (x,y)

• origin case (0,0,1)

• for (xh, yh, h), at least one of triple must be nonzero,

i.e. (0,0,0) is not allowed

• if (x,y,0) called point at infinity

For Translation, Rotation, Scaling

1

y

x

100

t10

t01

s

y

x

y

x

1

y

x

100

0cossin

0sin-cos

s

y

x

1

y

x

100

0s0

00s

s

y

x

y

xTranslation: Scaling:

Rotation:

Page 12: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Homogeneous Coordinates (4/4)

with the H.G.C we can perform uniform scaling with scaling factor S

• when S>1 : Uniform compression

0<S<1 : Uniform enlargement

with the H.G.C the transformation matrix be 3x3,

i.e. m, n : translation factor

s : scaling factorT

a c m

b d n

0 0 s

1

s

y

s

x

s

y

x

s00

010

001

1

y

x

Page 13: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Composite Transformation (1/2)

Translation If two successive translation factor (tx1, ty1) and (tx2, ty2) are

applied to a coordinate point P• then

• ex)

i.e,

Two successive Translation are additive

1 0 t

0 1 t

0 0 1

1 0 t

0 1 t

0 0 1

1 0 t t

0 1 t t

0 0 1

x2

y2

x1

y1

x1 x2

y1 y2

P' T (t ,t ) {T (t ,t ) P} {T (t ,t ) T (t ,t )} P2 X2 y2 1 x1 y1 2 X2 y2 1 x1 y1

T(t ,t ) T(t ,t ) T(t t ,t t )x2 y2 x1 y1 x1 x2 y1 y2

Page 14: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Composite Transformation (2/2)

Rotation

two successive rotation

two successive rotations are also additive

Scaling

successive scalings are multiplicative

P)}R(){R(P}){R()R( P 1212 )R()R()R( 2121

P' R( ) P1 2

,

100

0ss0

00ss

100

0s0

00s

100

0s0

00s

y2y1

x2x1

y1

x1

y2

x2

S(S ,S ) S(S ,S ) S(S S ,S S )X2 Y2 X1 Y1 X1 X2 Y1 Y2

Page 15: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

General pivot-point Rotation (1/2)

Rotation about arbitrary point p (xr, yr) step1) Translate P to origin step2) Rotation about origin step3) Retranslation to position P.

Page 16: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

General pivot-point Rotation (2/2)

Composite transformation matrix

x

y

x

y

x y

y x

r

r

r

r

r r

r r

cos sin

sin cos

cos sin ( cos ) sin

sin cos ( cos ) sin

),y,R(x),-yT(-x)R()y,T(x rrrrrr

Page 17: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

General Fixed-Point Scaling

Scaling about arbitrary point p(xr, yr) step 1) translate p to origin

step 2) scaling about origin

step 3) Retranslate to position P

x

y

s

s

x

y

s x s

s y s

r

r

x

y

r

r

x r x

y r y

( )

( ) )ss,y,S(x),-yT(-x)sS(s)y,T(x yxffffyxff ,,

Page 18: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

General Scaling Directions

Normal scaling performed along x and y directions To accomplish the scaling for arbitrary direction without

changing object orientation• Rotate direction by

• Perform scaling

• Rerotate by

R S S S R

s s s s

s s s s

( ) ( , ) ( )

cos sin ( )cos sin

( )cos sin sin cos

Page 19: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Concatenation properties.

Matrix multiplication : associative• ex) For three matrices A, B and C

• Translation or Rotation : additive property commutative

• scaling : multiplicative property commutative

However, Translation and Rotation : non commutative• order of transformation matrix multiplication is important

A B C (A B) C A (B C)

ABBA

Page 20: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Other Transformations : Reflection (1/4)

Reflection : produce a mirror images of an object We need axis of reflection rotating the objects 180about reflection axis Reflection about line y=0, the x axis.

• x coordinate values are unchanged

• the transformation matrix

R

1 0 0

0 1 0

0 0 1X

Page 21: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Reflection (2/4)

Reflection about line x=0, the y axis• y coordinate values are unchanged

Reflection about origin• both x and y values are changed

R

1 0 0

0 1 0

0 0 1y

100

010

001

R0

Page 22: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Reflection (3/4)

Reflection through arbitrary point P (xr, yr)• translate point p to origin : Tr

• perform reflection about origin : Ro

• retranslate to original position : Tr

Reflection through y=x line

100

y10

x01

100

010

001

100

y10

x01

r

r

r

r

0 1 0

1 0 0

0 0 1

Page 23: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Reflection (4/4)

Reflection through an arbitrary line y=Lx+b

Page 24: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Other Transformations: Shear (1/3)

Shear : distorts the shape of an object Shearing (slide over) can be done either x or y direction x direction shearing by shearing factor shx

• point is shifted by horizontally by an amount proportional to its distance from x-axis

• y values are unchanged

Page 25: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Shear (2/3)

• So,

• Transformed positions

y direction shearing

x-direction shearing relative to line

T

1 sh 0

0 1 0

0 0 1shx

x

x' x sh yx , y' y

T

1 0 0

sh 1 0

0 0 1shy y

y yref

1 sh sh y

0 1 0

0 0 1

x x ref

x' x sh (y y )

y' yx ref

Page 26: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Shear (3/3)

y-direction shearing relative to linex xref

1 0 0

sh 1 sh x

0 0 1y y ref

x' x

y' y sh (x x )y ref

Page 27: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Properties of Transformation matrix (1/5)

Properties of Rotation matrix case1) rotate point p p’ c.c.w direction by

• case2) rotate point p’ p c.w. direction by

• same result to rotate ( ) to c.c.w direction

• is a Inverse matrix of

• i.e,

P' P Rx

y

cos sin

sin cos(A)

det[R ] cos sin 12 2

R( )cos(- ) - sin(-

sin(- ) cos(- )

cos sin

sin cos(B)

)

R R( ) ( )

I

R( ) R( )

Page 28: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Properties of Transformation matrix (2/5)

Examine (A) & (B)

then Inverse matrix = Transpose matrix of R

I.e,

So, the inverse of the general rotation matrix [R]

is its transpose Def) An nxn matrix A is an orthogonal matrix if Def) Every 2x2 orthogonal matrix R with

det[R]=1 is pure rotation matrix

R RT

cos sin

sin cos

A AT

Page 29: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Properties of Transformation matrix (3/5)

Properties of Reflection matrix Determinant of Reflection matrix = -1 Def) Every 2x2 orthogonal matrix R with det(R) = -1

is a pure reflection matrix If two pure reflections about line passing through the origin a

re applied successively, the result is a pure rotation about the origin

• ex) reflection through x axis (RX)

and reflection through y axis (RY)

Page 30: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Properties of Transformation matrix (4/5)

Rigid-body Transformation concept : moves object without deformation

• i.e, perpendicular lines transformed as perpendicular lines

or unit square remains a unit square

Translation, rotation, or combination of both

Page 31: Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

Properties of Transformation matrix (5/5)

Affine Transformation Concepts :

• parallel lines transformed as parallel lines

• finite points maps to finite points

• but not length and angle

• Rot, Trans, Ref preserves angle and length

• Tra, Rot, Sca, Ref, and Shr or combination of those