graphics graphics lab @ korea university cgvr.korea.ac.kr 2d geometric transformations...

Graphics

cgvr.korea.ac.kr Graphics Lab @ Korea University

2D Geometric Transformations

고려대학교 컴퓨터 그래픽스 연구실

cgvr.korea.ac.kr

CGVR

Graphics Lab @ Korea University

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

cgvr.korea.ac.kr

CGVR


Geometric Transformation

Definition 물체의 좌표를 바꾸는 것 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

cgvr.korea.ac.kr

CGVR


Example: 2D Geometric Transformation

ModelingCoordinates

World Coordinates

cgvr.korea.ac.kr

CGVR


Example: 2D Scaling

ModelingCoordinates

World Coordinates

Scale(0.3, 0.3)

cgvr.korea.ac.kr

CGVR


Example: 2D Rotation

ModelingCoordinates

Scale(0.3, 0.3)Rotate(-90)

World Coordinates

cgvr.korea.ac.kr

CGVR


Example: 2D Translation

ModelingCoordinates

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

World Coordinates

cgvr.korea.ac.kr

CGVR



ModelingCoordinates

World Coordinates

Again?

cgvr.korea.ac.kr

CGVR



ModelingCoordinates

World Coordinates

ScaleTranslate

ScaleRotate

Translate

cgvr.korea.ac.kr

CGVR


Basic 2D Transformations

Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy

sinθy-cosθxx cosθysinθyy

yhxxx xhyyy

cgvr.korea.ac.kr

CGVR



Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy

Transformationscan be combined

(with simple algebra)

cgvr.korea.ac.kr

CGVR



Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy

syy

sxxx

y

cgvr.korea.ac.kr

CGVR


)cossy)(ysinsx)((xy

)sinsy)(ycossx)((xx


Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy

cgvr.korea.ac.kr

CGVR


ty)cossy)(ysinsx)((xy

tx)sinsy)(ycossx)((xx


Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy

cgvr.korea.ac.kr

CGVR



Translation

Scale

Rotation

Shear

txxx tyyy

sxxx syyy


yhxxx xhyyy

ty)cossy)(ysinsx)((xy

tx)sinsy)(ycossx)((xx

cgvr.korea.ac.kr

CGVR


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

cgvr.korea.ac.kr

CGVR


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

cgvr.korea.ac.kr

CGVR


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

cgvr.korea.ac.kr

CGVR


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

cgvr.korea.ac.kr

CGVR


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

cgvr.korea.ac.kr

CGVR


2×2 Matrices


2D Translation

txxx tyyy NO!!

Only linear 2D transformationscan be Represented with 2x2 matrix

cgvr.korea.ac.kr

CGVR


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

cgvr.korea.ac.kr

CGVR



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

cgvr.korea.ac.kr

CGVR


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

cgvr.korea.ac.kr

CGVR


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

cgvr.korea.ac.kr

CGVR


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

cgvr.korea.ac.kr

CGVR


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

cgvr.korea.ac.kr

CGVR


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

cgvr.korea.ac.kr

CGVR


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)

cgvr.korea.ac.kr

CGVR


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)

cgvr.korea.ac.kr

CGVR


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 ,,,, ,,

cgvr.korea.ac.kr

CGVR


Reflection

Reflection with respect to the axis • x 축에 대한 반사 • y 축에 대한 반사 • xy 축

( 원점 ) 에 대한

100

010

001

100

010

001

100

010

001

y 축에 대한 반사x 축에 대한 반사 원점에 대한 반사

x

y 1

32

1’

3’2’x

y1

32

1’

3’ 2

x

y

3

1’

3’ 2

1

2

cgvr.korea.ac.kr

CGVR


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 에 대한 반사

x

y 1

32

1’

3’

2’

x

y

x

y

x

y

cgvr.korea.ac.kr

CGVR


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 축으로 밀림 (Shx=2)

선분에 대한 밀림(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)

cgvr.korea.ac.kr

CGVR


Shear

Transformed to a shifted parallelogram(X = Xref)

x’ = x, y’ = shy · (x-xref) + y

100

1

001

refyy xshsh

선분에 대한 밀림(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)

graphics graphics lab @ korea university cgvr.korea.ac.kr 2d geometric transformations...

Documents