Chi-Cheng Lin, Winona State University

CS430 Computer Graphics

Transformations of Objects – 2D Part I

2

Topics

Matrix Prelim Introduction Fundamental Transformations Affine Transformations More Affine Transformations Inverse of Affine Transformations Composition of Affine

Transformations

3

Matrix Prelim Matrix: rectangle array of numbers A: nm matrix n rows, m columns

A vector in nD-space an n1 matrix

nmn

m

aa

a

aaa

1

21

11211

.:

.:

.

...

1

21

11

na

a

a

:

:

4

Matrix Multiplication

A: nm matrix B: mp matrixC = A B : np matrix, where

Ai =i th row vector of A

Bj =j th column vector of B

cij = Ai • Bj

A B = B A ?? A (B + C )= A B + A C ??

m

ssjisij bac

1

5

Identity Matrix

Identity matrix I

otherwise,

,

.:

.:

.

...

I

0

1

10

1

10

001

jiij

6

Determinant of Matrix Determinant of a 22 matrix A:

Determinant of an nn matrix A:

where A1i = determinant of an (n-1)(n-1) matrix obtained by deleting the first row and i th column from A

Find the determinant of a 33 matrix! (p.824)

n

jj

jj Aa

11

11 1)(detA

bcaddb

ca

AA det

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Transpose and Inverse of Matrix

Transpose of an nm matrix A is an mn matrix B B = A T, i.e., bij = aji Transpose of an n1 column vector is

a 1n row vector Inverse of an nn matrix A is an

nn matrix B B = A-1

A B = B A = I

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Introduction

Transformation: cornerstone of CG Change the position, size,

orientation of objects We will focus on linear

transformation Like other graphics systems,

transformation is also central to OpenGL

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OpenGL Graphics Pipeline OpenGL pipeline

ExampleglBegin(GL_LINES);

glVertex3f(…) // send P1 through the pipelineglVertex3f(…) // send P2 through the pipelineglVertex3f(…) // send P3 through the pipeline

glEnd();

CT Process& display

P1, P2, … Q1, Q2, …

P1

P2 Q1

Q2

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Transforming Points and Objects

Transformation alters each point P in space into a new point Q by means of formula or algorithmP is mapped to QQ is the image of P under the mapping T

Fundamental transformationsTranslateScaleRotate

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Fundamental Transformations - Translation

TranslationChange the position of an object(Qx, Qy) = (Px + dx, Py + dy)

orQ = P + T, where T = (dx, dy)

P2

P1

Q1

dx

dy x

yQ2

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Translation Example

Example

dx= 3dy=-4

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Fundamental Transformations - Scaling Scaling

Change the size of an object(Qx, Qy) = (Sx Px, Sy Py)

or or

Actually it is scaling about the origin

y

x

y

x

y

x

P

P

S

S

Q

Q

0

0

Sx=3

Sy=2

x

y

y

x

y

x

P

P

Q

QS

Sx=3

Sy=2

xX

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Scaling Example

Example

Sx=1/2Sy=1/4

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Fundamental Transformations - Rotation

RotationChange the orientation of an object(Qx, Qy) = (Pxcos -Pysin, Pxsin +

Pycos)

or or

Derivation of rotation mapping?

y

x

y

x

P

P

Q

Q

cossin

sin-cos

P

Q

x

y

y

x

y

x

P

P

Q

QR

rr

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Fundamental Transformation -Rotation

Rotation (cont’d)Rotation is performed about the

origin

x

y

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Rotation Example

Example

= 45o

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Affine Transformations Questions

How do we scale/rotate about any point?How do we combine transformations?

Problem of composing transformationsTranslation: Q = T + P additionScaling: Q = S P multiplicationRotation: Q = R P multiplication

Solution: homogeneous coordinatesRepresent a 2D point (x, y) as (x, y, 1) A vector (a, b) = (a, b, 0)

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Affine Transformations

Coordinates of point Q are linear combinations of those of point P:

Translation, scaling, and rotation are affine transformations (so are their combinations)

11232221

131211

mPmPm

mPmPm

Q

Q

yx

yx

y

x

11001232221

131211

y

x

y

x

P

P

mmm

mmm

Q

Q

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Affine Transformations Translation, scaling, and rotation are affine

transformations Succession of affine transformation is an

affine transformation Affine transformation from vector V to

vector W :

The third row of the transformation matrix is always (0, 0, 1)

01000232221

131211

y

x

y

x

V

V

mmm

mmm

W

W

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Fundamental Transformations Revisit

Translation(Qx, Qy) = (Px + dx, Py + dy)

1100

10

01

1y

x

y

x

y

x

P

P

d

d

Q

Q

100

10

01

y

x

yxyx d

d

ddPddQ ),(,),( T T

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Fundamental Transformations Revisit

Scaling(Qx, Qy) = (Sx Px, Sy Py)

1100

00

00

1y

x

y

x

y

x

P

P

S

S

Q

Q

100

00

00

x

x

yxyx S

S

SSPSSQ ),(,),( S S

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Fundamental Transformations Revisit

Rotation(Qx, Qy) = (Pxcos -Pysin, Pxsin +

Pycos)

1100

0

0

1y

x

y

x

P

P

Q

Q

cossin

sincos

100

0

0

cossin

sincos

)(,)( R R PQ

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More Affine Transformations - Reflection

Reflection – special case of scalingSx < 0 reflection about the y-axis

Sy < 0 reflection about the x-axis

Sx=3

Sy=2

y

Sx=-0.5

Sy=1

Sx=3

Sy=-2

x

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More Affine Transformations - Reflection

Pure reflectionEach of the scale factors is either +1 or –1Examples

T(Px, Py) = (-Px, Py) flip horizontally about y-axis

T(Px, Py) = (Px, -Py) flip vertically about x-axis

T(Px, Py) = (-Px, -Py) ?

Uniform scalingSx = Sy = S, where |S| is magnification factor

Differential scalingSx Sy

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More Affine Transformations - Shearing ShearX

x-coordinate is translated by an amount that increases linearly with y

y-coordinate is unaffectedCan be used to make italic

letters out of regular ones(Qx, Qy) = (Px + hPy, Py)

PhQP

Ph

Q

Q

xy

x

y

x

Sh )(

1100

010

01

1

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More Affine Transformations - Shearing

ShearYy-coordinate is translated by an amount

that increases linearly with xx-coordinate is unaffected(Qx, Qy) = (Px, gPx + Py)

PgQP

P

gQ

Q

yy

x

y

x

Sh )(

1100

01

001

1

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Inverse of Affine Transformations If Q =MP, then P =M–1Q, where M–1 is the

inverse of M (i.e., M M–1 = I )

where det M = m11m22 - m12m21

Inverse of an affine transformation is an affine transformation!

100232221

131211

mmm

mmm

M

100

1132111231121

2312221312221 )(

)(

detmmmmmm

mmmmmm

MM

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Inverse of Affine Transformations

Translation: T -1(dx, dy) = T(-dx, -dy)

Scaling: S -1(Sx, Sy) = S(1/Sx, 1/Sy) Rotation: R -1() = R(-) ShearX: ShX -1(h) = ShX (-h)

ShearY: Shy -1(g) = Shy(-g)

Verify that those matrices are correct If M = I, what transformation is it?

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Composition of Affine Transformations

Composing (concatenating) transformationsProcess of applying several transformations in

succession to form one overall transformation Composition of affine transformations is

affine If a sequence of transformations are

represented by matrices M1, M2, …, Mn, then a point P is transformed to a point Q = (Mn((M2(M1 P)))

= (Mn M2M1)P

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Composition of Affine Transformations

Example: Build a transformation matrix thatRotates 45 degreesScales in x by 1.5 and in y by –2Translates by (3, 5)Find the transformed point of P = (1, 2)Solution:

Q = M P = (1.94, 0.758, 1)

100

541414141

3061061

100

04545

04545

100

020

0051

100

510

301

..

..

)cos()sin(

)sin()cos(.

M

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Applications of Composition

Rotate about an arbitrary point (x, y)Translate by (-x, -y)Rotate about the origin through angle

Translate by (x, y)Transformation matrix

M = T(x, y)R()T(-x, -y)

100

yyx

xyx

)cos()sin()cos()sin(

)sin()cos()sin()cos(

12

3

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Applications of Composition

Scale about an arbitrary point (x, y)Translate by (-x, -y)Scale in x by Sx and in y by Sy Translate by (x, y) Transformation matrix M

= T(x, y)S(Sx, Sy)T(-x, -y) = ??

Window-to-viewport transformation is also a good example of composition of affine transformations (why?)

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Applications of Composition

Reflection about an axis of reflection that makes an angel of with the x axisRotate by -Reflect about x-axis

(i.e., scale in x by 1 and in y by –1)

Rotate by

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Applications of Composition Reflection about an axis of

reflection that makes an angel of with the x axisTranslation matrix M

= R()S(1, -1)R(-)Let c = cos(-) and s = sin(-),

100

022

022

100

02

02

100

0

0

100

010

001

100

0

022

22

)cos()sin(

)sin()cos(

cscs

cssc

cs

sc

cs

sc

M

Check it out!