Sun-Jeong Kim

2D Geometric Transformations

9th Week, 2008

Computer Graphics2

Geometric Transformations

DefinitionTo change the geometric description of an object like its position, orientation, or size

Why to need geometric transformations in CGTo reposition or resize objects

To convert a world-coordinates scene description to a display for an output device

Computer-aided design and computer animation

Modeling transformations

Computer Graphics3

Basic 2D Geometric Transformations

TranslationTranslation

RotationRotation

ScalingScaling

y

x

tyy

txx

+=′+=′

θθθθ

cossin

sincos

yxy

yxx

+=′−=′

y

x

syy

sxx

⋅=′⋅=′

x

y

x

y

x

y

x

y

x

y

x

y

Computer Graphics4

2D Translation

Adding offsets to the coordinates of a pointTranslation vectorTranslation vector (or shift vector): (tx, ty)

Rigid-body transformationMoving objects without deformation

y

x

tyy

txx

+=′+=′

class wcPt2D {public:

float x, y;};

void translatePolygon(wcPt2D *verts, int nVerts, float tx, float ty){

for(int k=0; k<nVerts; k++) {verts[k].x = verts[k].x + tx;verts[k].y = verts[k].y + ty;

}}

Computer Graphics5

2D Rotation (1)

Rotating all points of an objectA rotation axisrotation axis and a rotation anglerotation angle

A rotation point (pivot pointpivot point)

θθθφρθφρθθθφρθφρ

cossincossinsincos

sincossinsincoscos

yxy

yxx

+=+=′−=−=′

( )( )φθρ

φθρφρφρ

+=′+=′

==

sin

cos

sin

cos

y

x

y

x

Computer Graphics6

2D Rotation (2)

Rotating all points of an objectA rotation axisrotation axis and a rotation anglerotation angle

A rotation point (pivot pointpivot point)θθθθ

cossin

sincos

yxy

yxx

+=′−=′

class wcPt2D {public:

float x, y;};

void rotatePolygon(wcPt2D *verts, int nVerts, wcPt2D pivPt, double theta){

for(int k=0; k<nVerts; k++) {float rx = pivPt.x + (verts[k].x - pivPt.x) * cos(theta)

- (verts[k].y - pivPt.y) * sin(theta);float ry = pivPt.y + (verts[k].x - pivPt.x) * sin(theta)

+ (verts[k].y - pivPt.y) * cos(theta);verts[k].x = rx;verts[k].y = ry;

}}

Computer Graphics7

2D Scaling (1)

To alter the size of an objectMultiplying object positions (x, y) by scaling factorsscaling factors sx

and sy

Uniform scalingUniform scaling: sx = sy

Differential scaling (NonuniformNonuniform scalingscaling): sx≠sy

Fixed pointFixed point

y

x

syy

sxx

⋅=′⋅=′

( )( ) yff

xff

syyyy

sxxxx

⋅−+=′

⋅−+=′ ( )( )yfy

xfx

sysyy

sxsxx

−+⋅=′

−+⋅=′

1

1

Constants

Computer Graphics8

2D Scaling (2)

To alter the size of an objectMultiplying object positions (x, y) by scaling factorsscaling factors sx

and sy

Fixed pointFixed point

class wcPt2D {public:

float x, y;};

void scalePolygon(wcPt2D *verts, int nVerts, wcPt2D fixedPt, float sx, float sy)

{for(int k=0; k<nVerts; k++) {

verts[k].x = verts[k].x * sx + fixedPt.x * (1 – sx);verts[k].y = verts[k].y * sy + fixedPt.y * (1 – sy);

}}

( )( )yfy

xfx

sysyy

sxsxx

−+⋅=′

−+⋅=′

1

1

Computer Graphics9

Matrix Representations (1)

Translation

Rotation

Scaling

y

x

tyy

txx

+=′+=′

θθθθ

cossin

sincos

yxy

yxx

+=′−=′

y

x

syy

sxx

⋅=′⋅=′

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡′′

y

x

t

t

y

x

y

x

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡′′

y

x

y

x

θθθθ

cossin

sincos

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡′′

y

x

s

s

y

x

y

x

0

0

Transformation MatrixTransformation Matrix

Computer Graphics10

Matrix Representations (2)

Convenient and efficient way to represent a sequence of geometric transformations

Mi: transformation matrix

Ex) rotation scaling

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡′′

y

x

s

s

y

x

y

x

θθθθ

cossin

sincos

0

0

PMMMP 12Ln=′

Computer Graphics11

Matrix Representations (3)

How about ‘translation’?

Ex) rotation scaling translation:

Ex) translation rotation scaling: ???

to eliminate the matrix addition

Homogeneous coordinatesHomogeneous coordinates

21 MPMP +=′

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡′′

y

x

y

x

t

t

y

x

s

s

y

x

θθθθ

cossin

sincos

0

0

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡′′

y

x

t

t

y

x

y

x

10

01

Computer Graphics12

Homogeneous Coordinates (1)

Expanding each 2D coordinate-position representation (x, y) to 3-element representation (xh, yh, h)

Homogeneous parameterHomogeneous parameter: h

(x, y, w) (x/w, y/w, 1)

Ex) (2, 1, 1) = (4, 2, 2) = (6, 3, 3)

Cf. (x, y, 0) a point at infinite = a vector

h

yy

h

xx hh == ,

1 2

1

2

x

y

Computer Graphics13

Homogeneous Coordinates (2)

Translation

Rotation

Scaling

y

x

tyy

txx

+=′+=′

θθθθ

cossin

sincos

yxy

yxx

+=′−=′

y

x

syy

sxx

⋅=′⋅=′

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡′′

1100

10

01

1

y

x

t

t

y

x

y

x

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡′′

1100

0cossin

0sincos

1

y

x

y

x

θθθθ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡′′

1100

00

00

1

y

x

s

s

y

x

y

x

Computer Graphics14

Inverse Transformations

Translation

Rotation

Scaling

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100

10

01

y

x

t

t

T

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

100

0cossin

0sincos

θθθθ

R

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100

00

00

y

x

s

s

S

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−

=−

100

10

011

y

x

t

t

T

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=−

100

0cossin

0sincos1 θθ

θθR

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=−

100

010

0011

y

x

s

s

S

Computer Graphics15

2D Composite Transformations

A sequence of transformations multiplication of transformation matrices

ConcatenationConcatenation (composition)

Efficiency with premultiplication

Ex) translation rotation scaling:MPP

PMMMP

=′=′ 12Ln

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡′′

1100

10

01

100

0cossin

0sincos

100

00

00

1

y

x

t

t

s

s

y

x

y

x

y

x

θθθθ

Computer Graphics16

General 2D Pivot-Point Rotation

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−+−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100

sin)cos1(cossin

sin)cos1(sincos

100

10

01

100

0cossin

0sincos

100

10

01

θθθθθθθθ

θθθθ

rr

rr

r

r

r

r

xy

yx

y

x

y

x

( ) ( ) ( ) ( )θθ ,,,, rrrrrr yxyxyx RTRT =−−⋅⋅

Translate Rotate Translate

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

Computer Graphics17

General 2D 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 ssyxyxssyx ,,,, ,, STST =−−⋅⋅

Computer Graphics18

Reflection (1)

Reflection with respect to the axis

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

100

010

001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

100

010

001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−

100

010

001

Reflectionabout the x axis

x

y 1

32

1’

3’2’x

y1

32

1’

3’ 2’

x

y

3

1’

3’ 2’

1

2

Reflectionabout the y axis

Reflectionrelative to the origin

Computer Graphics19

Reflection (2)

Reflection with respect to a line

Clockwise rotation of 45° reflection about the x axis counterclockwise rotation of 45°

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100

001

010 y=x

x

y

x

y 1

32

1’

3’2’x

y

x

y

Computer Graphics20

Shear (1)

Distorting the shape of an abjectA unit square parallelogram

A unit square shifted parallelogram

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100

010

01 xh

x-direction shear with hx=2

x

y

x

y

(0,0) (1,0)

(1,1)(0,1)

(0,0) (1,0)

(3,1)(2,1)

yy

yhxx x

=′⋅+=′

Shear with hx=1/2, yref=–1

x

y

x

y

(0,0) (1,0)

(1,1)(0,1)

(1/2,0)

(3/2,0)

(2,1)(1,1)

(0,-1)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ ⋅−

100

010

1 refxx yhh ( )yy

yyhxx refx

=′

−+=′