Hearn and Baker 2 D transformations

2D Geometric Transformations

Taher S. Vijay Computer Academy

August 2011

Taher S. - VCA- 2011

Contents

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


Example: 2D Geometric Transformation

ModelingCoordinates

World Coordinates


Example: 2D Scaling

ModelingCoordinates

World Coordinates

Scale(0.3, 0.3)


Example: 2D Rotation

ModelingCoordinates

Scale(0.3, 0.3)Rotate(-90)

World Coordinates


Example: 2D Translation

ModelingCoordinates

Scale(0.3, 0.3)Rotate(-90)Translate(5, 3)

World Coordinates



ModelingCoordinates

World Coordinates

Again?



ModelingCoordinates

World Coordinates

ScaleTranslate

ScaleRotate

Translate


Basic 2D Transformations

Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy

sinθy-cosθxx cosθysinθyy

yhxxx xhyyy



Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy

Transformationscan be combined

(with simple algebra)



Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy

syy

sxxx

y


)cossy)(ysinsx)((xy

)sinsy)(ycossx)((xx


Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy


ty)cossy)(ysinsx)((xy

tx)sinsy)(ycossx)((xx


Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy



Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy

ty)cossy)(ysinsx)((xy

tx)sinsy)(ycossx)((xx


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 Matrices

What types of transformations can be represented with a 2×2 matrix?

2D Identity

2D Scaling

yy

xx

ysyy

xsxx

y

x

y

x

10

01

y

x

sy

sx

y

x

0

0


2×2 Matrices


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 Matrices


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 Matrices


2D Translation

txxx tyyy

y

x

ty

tx

y

x

0

0

Careful!


Homogeneous Coordinates

Add a 3rd coordinate to every 2D point (x, y, w) represents a point at location (x/w, y/w) (x, y, 0) represents a point at infinity (0, 0, 0) Is not allowed

1 2

1

2

x

y

(2, 1, 1) or (4, 2, 2) or (6, 3, 3)

Convenient Coordinate System to

Represent Many Useful Transformations


2D Translation

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

txxx tyyy

1100

10

01

1

y

x

ty

tx

y

x



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


Linear Transformations

Linear transformations are combinations of … Scale Rotation Shear, and Mirror

Properties of linear transformations Satisfies: Origin maps to origin Lines map to lines Parallel lines remain parallel Ratios are preserved Closed under composition

)()()( 22112211 pTspTspspsT

w

y

x

dc

ba

w

y

x

100

0

0


Affine Transformations

Affine transformations are combinations of Linear transformations, and Translations

Properties of affine transformations Origin does not map to origin Lines map to lines Parallel lines remain parallel Ratios are preserved Closed under composition

w

y

x

fed

cba

w

y

x

100


Projective Transformations

Projective transformations… Affine transformations, and Projective warps

Properties of projective transformations Origin does not map to origin Lines map to lines Parallel lines do not necessarily remain parallel Ratios are not preserved Closed under composition

w

y

x

ihg

fed

cba

w

y

x


Matrix Composition

Transformations can be combined by matrix multiplication

Efficiency with premultiplication Matrix multiplication is associative

w

y

x

1 0 0

0sy0

00sx

1 0 0

0cosθinθ

0 sinθ- θcos

100

10

01

sty

tx

w

y

x

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

p)))S(R(T(p pS)RT(p


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 ,,,, ,,


Reflection

Reflection with respect to the axis • x • y • xy

100

010

001

100

010

001

100

010

001

yx

x

y 1

32

1’

3’2’x

y1

32

1’

3’ 2

x

y

3

1’

3’ 2

1

2

xy


Reflection with respect to a Line

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

Reflection

100

001

010

y=x

x

y 1

32

1’

3’

2’

x

y

x

y

x

y


Shear

Converted to a parallelogram

x’ = x + shx · y, y’ = y

Transformed to a shifted parallelogram(Y = Yref)

x’ = x + shx · (y-yref), y’ = y

100

010

01 xsh

100

010

1 refxx yshsh

x (Shx=2)

(Shx=1/2, yref=-1)

x

y

x

y

x

y

x

y

(0,0) (1,0)

(1,1)(0,1)

(0,0) (1,0)

(1,1)(0,1)

(0,0) (1,0)

(3,1)(2,1)

(1/2,0)(3/2,0)

(2,1)(1,1)

(0,-1)


Shear

Transformed to a shifted parallelogram(X = Xref)

x’ = x, y’ = shy · (x-xref) + y

100

1

001

refyy xshsh

(Shy=1/2, xref=-1)

x

y

x

y

(-1,0)(0,0) (1,0)

(1,1)(0,1)(0,1/2) (1,1)

(1,2)(0,3/2)

Technology

Hearn and Baker 2 D transformations