Chapter 4

2D Geometric Transformations

Agenda• Definition & Motivation• 2D Geometric Transformation

– Translation– Rotation– Scaling

• Matrix Representation• Homogeneous Coordinates• Matrix Composition• Composite Transformations

– Pivot-Point Rotation– General Fixed-Point Scaling – Reflection and Shearing– Transformations Between Coordinate Systems

Geometric Transformation

• Definition – Translation, Rotation, Scaling

• Motivation – Why do we need geometric

transformations in CG?– As a viewing aid– As a modeling tool– As an image manipulation tool

Basic 2D Transformations• Translation

– –

• Scale– –

• Rotation– –

• Shear– –

txxx tyyy

sxxx syyy

sinθy-cosθxx

cosθysinθxy

yhxxx S

xhyyy S

Matrix NotationsTransformation

NameTransformation Matrix Notation

Inverse Transformation

Matrix

Translation T(a, b) T-1(a,b) = T(-a, -b)

Rotation R(θ) R-1(θ) = R(-θ)

Scaling S(sx, sy) S-1 (sx, sy) =()

Reflection About x-axis

Mx(a, b) Mx-1 (a, b) = Mx(a, -b)

Reflection About y-axis

My(a, b) My-1 (a, b) = My(-a, b)

Reflection About the origin

Mo(a, b) Mo-1 (a, b) = Mo(-a, -b)

Matrix Representation• Represent a 2D Transformation by a Matrix

• Apply the Transformation to a Point

y

x

dc

ba

y

x

dycxy

byaxx

dc

ba

TransformationMatrix Point

Matrix Representation• Transformations can be combined by matrix

multiplication

y

x

lk

ji

hg

fe

dc

ba

y

x

Matrices are a convenient and efficient way

to represent a sequence of transformations

TransformationMatrix

2×2 Matrix (2D Scaling)

2D Identity

2D Scaling

yy

xx

ysyy

xsxx

y

x

y

x

10

01

y

x

sy

sx

y

x

0

0

2×2 Matrix (2D Rotation & Shearing)

2D Rotation

2D Shearing

y

x

y

x

cossin

sincos

y

x

shy

shx

y

x

1

1

yxy

yxx

cossin

sincos

yxshyy

yshxxx

2×2 Matrix (2D Reflection)

2D Mirror over Y axis

2D Mirror over (0,0)

yy

xx

yy

xx

y

x

y

x

10

01

y

x

y

x

10

01

2×2 Matrix (2D Translation)• This transformation cannot be represented with

a 2×2 matrix?

2D Translation

txxx tyyy

NO!!

Only linear 2D transformationscan be Represented with 2x2 matrix

Homogeneous coordinates• Reason?

– Unification and extension of concepts– Best way to represent a sequence of

transformations

• 2D translation can be represented by a 3×3 matrix– Point represented with homogeneous coordinates

txxx tyyy

1

y

x

100

ty10

tx01

1

y

x

Basic 2D Translation

A translation moves all points of an object a fixed distance in a specified direction.

txxx

tyyy

1

y

x

100

ty10

tx01

1

y

x

Basic 2D RotationIt rotates points in the xy-Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system.

1

y

x

100

0cossin

0sincos

1

y

x

θ siny-θ cosxx

θ cosyθ sinxy

Basic 2D ScalingIt is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions.

1

y

x

100

0sy0

00sx

1

y

x

syyy sxxx

Basic 2D Shearing• A transformation that distorts the shape of an

object.• Two types of shearing transformations are

there about X values and about Y values

yshxxx xshy yy

1

y

x

100

01shy

0shx1

1

y

x

Basic 2D Transformations• Basic 2D transformations as 3x3 Matrices

1100

10

01

1

y

x

ty

tx

y

x

1100

0cossin

0sincos

1

y

x

y

x

1100

00

00

1

y

x

sy

sx

y

x

1100

01

01

1

y

x

shy

shx

y

x

Translate

Shear

Scale

Rotate

Matrix Composition• Transformations can be combined by matrix

multiplication

– Matrix multiplication is associative

w

y

x

1 0 0

0sy0

00sx

1 0 0

0cosθinθ

0 sinθ- θ

s

cos

100

ty10

tx01

w

y

x

p ty)T(tx, )R( sy)S(sx, p

p)))SRTp ((( pS)RTp (

Matrix Composition• Rotate by around arbitrary point (a,b)

• Scale by sx, sy around arbitrary point (a,b)

T(-a,-b))R(b)T(a,M θ

T(-a,-b)sy)S(sx,b)T(a,M

(a,b)

(a,b)

Pivot-Point Rotation

100

sin)cos1(cossin

sin)cos1(sincos

100

10

01

100

0cossin

0sincos

100

10

01

rr

rr

r

r

r

r

xy

yx

y

x

y

x

,,,, rrrrrr yxRyxTRyxT

Translate Rotate Translate

(xr,yr) (xr,yr) (xr,yr)(xr,yr)

General Fixed-Point Scaling

100

)1(0

)1(0

100

10

01

100

00

00

100

10

01

yfy

xfx

f

fx

f

f

sys

sxs

y

x

s

s

y

x

y

Translate Scale Translate

(xf,yf) (xf,yf) (xf,yf) (xf,yf)

yxffffyxff ssyxSyxTssSyxT ,,,, ,,

Translate-Rotate-Translate

Original position

Final position

1

y

x

1 0 0

10

01

1 0 0

0cosθinθ

0 sinθ- θcos

100

10

01

1

y-

x- r

rr

r

sy

x

y

x

Reflection• Reflection about x-axis (y=0)• Reflection about y-axis (x=0)• Refection about the origin (0, 0)• Reflection about the line y = x• Reflection about the line y = ax + b

Reflection about x-axis (y=0)

100

010

001

y 1

x

32

1’

3’2’

Reflection about y-axis (x=0)

100

010

001

x

y1

32

1’

3’ 2

Refection about the origin (0, 0)

100

010

001

x

y

3

1’

3’ 2

1

2

Reflection with respect to a Line

Clockwise rotation of 45 Reflection about the x axis Counterclockwise rotation of 45

Reflection about the line y = x

100

001

010

x

y 1

32

1’

3’2’x

y

x

y

x

y

Reflection about the line y = -x

100

001

010

1) Translation

2) Rotation

3) Reflection

4) Reverse Rotation

5) Reverse Translation

Reflection about the line y = ax+b

x

y