Lecture05 Transformation

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Com uter Gra hics

Chapter 5

Geometric Transformations

Somsak Walairacht, Computer Engineering, KMITL 1


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Outline

Basic Two-Dimensional Geometric Transformations Matrix Representations and Homogeneous

Inverse Transformations

wo-Dimensional Composite ransformations

Other Two-Dimensional Transformations

Raster Methods for Geometric Transformations

pen as er rans orma ons Transformations between Two-Dimensional

Coordinate Systems

201074410 / 13016218 Computer Graphics


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Outline (2)

Geometric Transformations in Three-DimensionalSpace

Rotation

Scaling -

Other Three-Dimensional Transformations

ransformations between hree-DimensionalCoordinate Systems

Affine Transformations



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Introduction

Operations that are applied to the geometricdescription of an object to change its position,orientation, or size are called geometrictransformations

Sometimes geometric-transformation operations arealso referred to as modeling transformations

A distinction between the two Modeling transformations are used to construct a scene or to

give the hierarchical description of a complex object that is

Geometric transformations describe how objects might movearound in a scene during an animation sequence or simplyto view them from another angle



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Basic Two-DimensionalGeometric ransformations

Available in all graphics packages are

Rotation

Other useful transformation routines are

e ec on Shearing



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wo-Dimensional Translation

To translate a two-dimensional position, we add translationdistances tx and ty to the original coordinates (x, y) to obtain thenew coordinate position (x, y)

Translation is a rigid-body transformation that moves objects

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w ou e orma on



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Polygon Translation

A polygon is translated similarly Adding a translation vector to the coordinate

position of each vertex and then regenerate

the polygon using the new set of vertex



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wo-Dimensional Rotation

A rotation transformation is generated by specifying arotation axis and a rotation angle

(xr, yr) called the rotation point (or pivot point),about which the object is to be rotated

A ositive value for the an le defines a

counterclockwise rotation about the pivot point



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wo-Dimensional Rotation (2)

Let r is the constant distance of the pointfrom the origin, angle is the original angular,

is the rotation angle



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Rotation about Arbitrary Point

Transformation equations for rotation of a pointabout any specified rotation position (xr, yr)

Rotations are ri id bod transformations that move

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objects without deformation



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wo-Dimensional Scaling

A simple two-dimensional scaling operation is performed bymultiplying object positions (x, y) by scaling factors sx and sy toproduce the transformed coordinates (x, y)

,



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Scaling by a Fixed Point

Coordinates for the fixed point, (xf, y

f), are often chosen at

some object position, such as its centroid Objects are now resized by scaling the distances between object

where the additive terms xf(1sx) andyf(1sy) are constants for all points in the object



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Matrix Representations andHomogeneous Coordinates

Many graphics applications involve sequencesof geometric transformations

An animation might require an object to be

translated and rotated at each increment of the

The viewing transformations involve sequences oftranslations and rotations to take us from the

original scene specification to the display on anoutput device



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Matrix M1 is a 2x2 array containing multiplicative, 2 -

containing translational terms

Multiplicative and translational terms can becom ne nto a s ng e matr x we expan t erepresentations to 3x3

A three-element re resentation x , , h , called

homogeneous coordinates, where the homogeneousparameter h is a nonzero value such that



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Two-Dimensional Translation Matrix

Rotation Matrix



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Inverse Transformations

Inverse translation matrix Translate in the opposite direction

Inverse rotation matrix

Inverse scaling matrix



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Two-Dimensional Compositeransformations

Forming products of transformation matrices is oftenreferred to as a concatenation, or composition, of

We do premultiply the column matrix by the matrices

representing any transformation sequence nce many pos t ons n a scene are typ ca y

transformed by the same sequence, it is moreefficient to first multiply the transformation matrices

to orm a sing e composite matrix



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Composite Two-Dimensionalranslations

If two successive translation vectors (t1x

, t1y

) and (t2x

, t2y

) areapplied to a 2-D coordinate position P, the final transformedlocation P is

The composite transformation matrix for this sequence oftranslations is



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Composite Two-DimensionalRotations

Two successive rotations applied to a point P

We can verify that two successive rotations are

additive:

e compos on ma r x



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Composite Two-DimensionalScalings

For two successive scaling operations in-

scaling matrix



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General Two-DimensionalPivot-Point Rotation

Graphics package provides only a rotatefunction with respect to the coordinate

To generate a 2-D rotation about any other

pivot point (xr, yr), follows the sequence ofrans a e-ro a e- rans a e opera ons1. Translate the object so that the pivot-point

position is moved to the coordinate origin

2. Rotate the object about the coordinate origin3. Translate the object so that the pivot point is

returned to its original position



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General Two-DimensionalPivot-Point Rotation (2)



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General Two-DimensionalFixed-Point Scaling

1. Translate the object so that the fixed point coincides with thecoordinate origin

2. Scale the object with respect to the coordinate origin3. se e nverse o e rans a on n s ep o re urn e

object to its original position



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General Two-DimensionalScaling Directions

To accomplish the scaling without changing theorientation of the object1. Performs a rotation so that the directions for s and s

coincide with the x and y axes2. Scaling transformation S(s1, s2) is applied

3. An opposite rotation to return points to their original



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Example of Scalingransformation

Turn a unit square into a parallelogram by stretchingit along the diagonal from (0, 0) to (1, 1)

Double its length with the scaling values s1=1 and s2=2

Rotate again to return the diagonal to its original orientation



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Matrix ConcatenationProperties

Multiplication of matrices is associative

Depending upon the order in which the transformations

by multiplying from left-to-right (premultiplying) - -

In OpenGL's convention: V'=B*(A*V), V'=(B*A)*V, C=B*A; V'=C*V;

In DX's convention: V'=(V*A)*B, C=A*B; V'=V*C;



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General Two-Dimensional Composite

Efficiency

. ,additions

Actually, only 4 multiplications and 4 additions

Once matrix is concatenated, it is maximum number of

Without concatenation, individual transformations would beapplied one at a time, and the number of calculations couldbe significantly increased



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Computational Efficiency (2)

Rotation calculations require trigonometricevaluations and several multiplications

consideration in rotation transformations

For small enough angles (less than 10), cosisapprox mate y . an s n as a va ue very c ose tothe value of in radians



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Computational Efficiency (3)

Composite transformations ofteninvolve inverse matrices

Operations are much simpler than direct

inverse matrix calculations Inverse translation matrix is obtained by

changing the signs of the translation

Inverse rotation matrix is obtained by

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Two-Dimensional Rigid-Bodyransformation

All angles and distances between coordinatepositions are unchanged by therans orma on

Upper-left 2x2 submatrix is an orthogonal

Two row vectors (rxx, rxy) and (ryx, ryy) (or the twocolumn vectors) form an orthogonal set of unit

vec ors Set of vectors is also referred to as an

orthonormal vector set



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Two-Dimensional Rigid-Bodyransformation (2)

Each vector has unit length

If these unit vectors are transformed by the rotation sub-matrix, then

Example,



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Constructing Two-DimensionalRotation Matrices

The orthogonal property of rotation matrices is useful for

constructing the matrix when we know the final orientation ofan object

,matrix within objects co-or system when knowing its orientationwithin overall word co-or system



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Other Two-Dimensionalransformations

Reflection



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Reflection

Reflection about the liney = 0 (the x axis) is

A reflection about theline x = 0 (the y axis)w

transformation matrixx r w

keeping y coordinates

the same



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Reflection (2)

Reflection relative tothe coordinate origin

Reflection axis as thediagonal line y = x



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Shear

A transformation that distorts the shape of an object The transformed shape appears as if the object were

slide over each other An x-direction shear relative to the x axis



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Shear (2)

x-direction shears relative to other reference lines

y-direction shear relative to the line x = xref



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Raster Methods for Geometricransformations

All bit settings in the rectangular area shown are copied as a block into

another part of the frame buffer

Rotate a two-dimensional object or pattern 90 counterclockwise byreversin the ixel values in each row of the arra then interchan in

rows and columns



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Raster Methods for Geometricransformations (2)

For array rotations that are not multiples of90, we need to do some extra processing

Similar methods to scale a block of pixels



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ransformations between Two-Dimensional Coordinate Systems

Computer-graphics applications involve coordinate

transformations from one reference frame to another duringvarious stages of scene processing

- -1. Translate so that the origin (x0, y0) of the xy system is moved to

the origin (0, 0) of the xy system

2. Rotate the x axis onto the x axis



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Transformations between 2-DCoordinate Systems (2)

Alternative method Specify a vector V that indicates the direction for the

Obtain the unit vector u along the x axis by applying

a 90 clockwise rotation to vector v

set of orthonormal vectors



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Geometric Transformations inhree-Dimensional Space

A position P=(x, y, z) in 3-D is translated to alocation P=(x, y, z) by adding translation distancestx, t , and tz



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3-Dimensional Rotation

-



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3-Dimensional Rotation (2)

Transformation equations for rotations about theother two coordinate axes can be obtained with acyclic permutation of the coordinate parameters x, y,and z

x y z x



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3-Dimensional Rotation (3)

x-axis rotation

y-axis rotation


h l


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Composite Three-Dimensionalransformations

A rotation matrix for any axis that does notcoincide with a coordinate axis

1. Translate the object so that the rotation axiscoincides with the parallel coordinate axis

.

3. Translate the object so that the rotation axis ismoved back to its ori inal osition


C i Th Di i l


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Composite Three-Dimensionalransformations (2)

A coordinate position P is transformed with thesequence

w ere


C it Th Di i l


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Rotate about an axis that is not parallel to one of

the coordinate axes

1. Translate the object so that the rotation axis passesthrough the coordinate origin

2. Rotate the object so that the axis of rotation coincides with

3. Perform the specified rotation about the selected coordinateaxis

original orientation5. Apply the inverse translation to bring the rotation axis back

to its original spatial position


C it Th Di i l


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C it Th Di i l


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The components of the rotation-axis vector

The unit rotation-axis vector u

Move the point P1 to the origin

x-axis rotation ets u into the xz lane

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y-axis rotation swings u around to the z axis


C it Th Di i l


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Rotate around the x axis to get u into

|u |=1 |u|=d

Rotation matrix


Composite Three Dimensional


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Resulting from the rotation aboutx axis is a vector labeled u

about the y axis is


Composite Three Dimensional


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The rotation axis is aligned with the positive z axis

Apply the specified rotation angle

To complete the required rotation about the givenaxis, we need to transform the rotation axis back to


Quaternion Methods for


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Quaternion Methods forhree-Dimensional Rotations

A more efficient method Require less storage space than 4x4 matrices Important in animations, which often require complicated motion

of an objectq=(a,b,c)

MR() = Rx-1() Ry

-1() Rz() Ry() Rx()


Other Three Dimensional


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Other Three-Dimensionalransformations

Scaling


Other Three Dimensional


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Other Three-Dimensional

ransformations (2)

Reflection

Shear



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Affine Transformations

An affine transformation is a form of coordinate transformation

av ng e genera proper es a para e nes are rans ormeinto parallel lines and finite points map to finite points

Examples

Conversion of coordinate descriptions from one reference system toanother

An affine transformation involving only translation, rotation, and

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,


OpenGL Geometric-


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OpenGL Geometric

ransformation Functions

Basic OpenGL Geometric Transformations glTranslate* (tx, ty, tz);

-. , . , .

glRotate* (theta, vx, vy, vz);

glRotatef (90.0, 0.0, 0.0, 1.0);

, ,

glScalef (2.0, -3.0, 1.0);

OpenGL Matrix Operations glMatrixMode (GL_MODELVIEW); glMatrixMode (GL_PROJECTION);



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Lecture05 Transformation

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