2d geometric transformations

Graphics

Graphics Lab @ Korea University

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2D Geometric Transformations

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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

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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

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Example: 2D Geometric Transformation

ModelingCoordinates

World Coordinates

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Example: 2D Scaling

ModelingCoordinates

World Coordinates

Scale(0.3, 0.3)

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Example: 2D Rotation

ModelingCoordinates

Scale(0.3, 0.3)Rotate(-90)

World Coordinates

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Example: 2D Translation

ModelingCoordinates

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

World Coordinates

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ModelingCoordinates

World Coordinates

Again?

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ModelingCoordinates

World Coordinates

ScaleTranslate

ScaleRotate

Translate

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Basic 2D Transformations

Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy

sinθy-cosθxx cosθysinθyy

yhxxx xhyyy

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Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy

Transformationscan be combined

(with simple algebra)

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Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy

syy

sxxx

y

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)cossy)(ysinsx)((xy

)sinsy)(ycossx)((xx


Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy

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ty)cossy)(ysinsx)((xy

tx)sinsy)(ycossx)((xx


Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy

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Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy

ty)cossy)(ysinsx)((xy

tx)sinsy)(ycossx)((xx

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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

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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

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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

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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

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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

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2×2 Matrices


2D Translation

txxx tyyy NO!!

Only linear 2D transformationscan be Represented with 2x2 matrix

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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

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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

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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

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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

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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)

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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

)

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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 ,,,, ,,

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Reflection

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

100

010

001

100

010

001

100

010

001

x

y 1

32

1’

3’2’x

y1

32

1’

3’ 2

x

y

3

1’

3’ 2

1

2

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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

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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

(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)

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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)

2d geometric transformations

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