Geometric Transformations

UBI 516 Advanced Computer GraphicsUBI 516 Advanced Computer Graphics

Aydın Öztürkozturk@ube.ege.edu.trhttp://ube.ege.edu.tr/~ozturk

Two Dimensional Geometric Transformations

Basic TransformationsBasic Transformations

Translation

P

P'

x

yyx tyytxx ','

y

x

t

t

y

x

y

xTP'P ,

'

',

TPP'

Basic Transformations

Rotation

x

y

y)(x,P θ

φ

)('

)('

rSiny

rCosx

)(),( rSinyrCosx

)()('

)()('

yCosxSiny

ySinxCosx

P′=(x′,y′)

r

Rotation (Cont.)

R.PP'

y

x

y

x

CosSin

SinCos

CosSin

SinCosR

'

'

Basic Transformations

Scaling

yx syysxx .',.'

PSP

'

.0

0

'

'

y

x

s

s

y

x

y

x

x

y

Basic Transformations

3x3 Matrix Representations

We can combine the multiplicative and translational terms for 2D transformations into a single matrix representation by expanding the 2x2 matrix representations to 3x3 matrices.

This allows us to express all transformation equations as matrix multiplications.

Homogeneous Coordinates

We represent each Cartesian coordinate position (x,y) with the homogeneous coordinate triple

where

),,( hyx hh

hy

yhx

x hh ,

Homogeneous Coordinates(cont.)

Thus, a general homogeneous coordinate representation can also be written as

For 2D transformations we choose h=1.

Each 2D position is represented with

homogeneous coordinates

),.,.( hyhxh

)1,,( yx

Translation in homogeneous coordinates

1

.

100

10

01

1

'

'

y

x

t

t

y

x

y

x

PTP ),(' yx tt

Rotation in homogeneous coordinates

1

.

100

0

0

1

'

'

y

x

CosSin

SinCos

y

x

PRP ).('

Scaling in homogeneous coordinates

1

.

100

00

00

1

'

'

y

x

s

s

y

x

y

x

PSP ).,(' yx ss

Composite Transformations:Translation

If two successive translation are applied to a point P, then the final transformed location P' is calculated as

100

10

01

100

10

01

.

100

10

01

21

21

1

1

2

2

yy

xx

y

x

y

x

tt

tt

t

t

t

t

PTPTTP ),(),(),('21211122 yyxxyxyx tttttttt

Composite Transformations:Rotation

100

0)()(

0)()(

100

0

0

.

100

0

0

2121

2121

11

11

22

22

CosSin

SinCos

CosSin

SinCos

CosSin

SinCos

PRP )(' 21

Composite Transformations:Scalings

100

00

00

100

00

00

100

00

00

21

21

1

1

2

2

yy

xx

y

x

y

x

ss

ss

s

s

s

s

),(),(),(21211122 yyxxyxyx ssssssss SSS

General Pivot Point Rotation

Steps:

-Translate the object so that the pivot point is moved to the coordinate origin.

-Rotate the object about the origin.

-Translate the object so that the pivot point is returned to its original position.

General Pivot Point Rotation(Cont.)

General Pivot Point Rotation(Cont.)

100

)1(

)1(

100

10

01

.

100

1

0

.

100

10

01

SinxCosyCosSin

SinyCosxSinCos

y

x

CosSin

SinCos

y

x

rr

rr

r

r

r

r

),,(),()(),( rrrrrr yxyxyx RTRT

General Fixed Point Scaling

Steps:

-Translate the object so that the fixed point coincides with the coordinate origin.

-Scale the object about the origin.

-Translate the object so that the pivot point is returned to its original position.

General Fixed Point Scaling

(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

f

y

x

f

f

sys

sxs

y

x

s

s

y

x

),,,(),(),(),( yxffffyxff ssyxyxssyx STST

Concatenation Properties

Matrix multiplication is associative

Transformation product is not commutative

)()( CBACBACBA

ABBA

OTHER TRANSFORMATIONS

Reflection

About x-axis About y-axis

x

y

1

2 3

2 3

1

100

010

001

100

010

001

OTHER TRANSFORMATIONS

x

y

1

2

3

2

1

3

100

010

001

Reflection about the origin

OTHER TRANSFORMATIONS

Reflection about the line y=x.

x

y2

2

1

3

100

001

010

y = x

2

3

OTHER TRANSFORMATIONS

100

010

01xhs

Shear

x-direction shearx x

y y

TRANSFORMATION BETWEEN COORDINATE SYSTEMS

Individual objects may be defined in their local cartesian reference system.

The local coordinates must be transformed to position the objects within the scene coordinate system.

TRANSFORMATION BETWEEN COORDINATE SYSTEMS(Cont.)

Steps for coordinate transformation

-Translate so that the origin (x0, y0 ) of the x′-y′ system is moved to the origin of the x-y system.

-Rotate the x′ axis on to the axis x.

TRANSFORMATION BETWEEN COORDINATE SYSTEMS(Cont.)

y

x

x′y′

θ

x0

y0

0

TRANSFORMATION BETWEEN COORDINATE SYSTEMS(Cont.)

y

x

x′

x0

y0

0

y′

θ

TRANSFORMATION BETWEEN COORDINATE SYSTEMS(Cont.)

y

xx′ x0

y0

0

y′

TRANSFORMATION BETWEEN COORDINATE SYSTEMS(Cont.)

),()(

100

0

0

)(

100

10

01

),(

00'',

0

0

00

yx

CosSin

SinCos

y

x

yx

yxxy

TRM

R

T

TRANSFORMATION BETWEEN COORDINATE SYSTEMS(Cont.)

An alternative method: -Specify a vector V that indicates the direction for the positive y′ axis. Let

-Obtain the unit vector u=(ux ,u y) along the x′ axis by rotating v 900 clockwise.

),( yx vvVV

v

TRANSFORMATION BETWEEN COORDINATE SYSTEMS(Cont.)

-Elements of any rotation matrix can be expressed as elements of orhogonal unit vectors. That is, the rotation matrix can be written as

100

0

0

yx

yx

vv

uu

R

TRANSFORMATION BETWEEN COORDINATE SYSTEMS(Cont.)

y

x

x′y′

x0

y0

0

V

Three Dimensional Geometric Transformations

P=(x,y,z)

x

3D Translation

P′=(x′,y′,z′)y

z

PTP

'

1

.

1000

100

010

001

1

'

'

'

z

y

x

t

t

t

z

y

x

z

y

x

3D Translation

3D Rotation(z-axis)

z-axis rotation

P=(x,y,z)

x

y

z

3D Rotation(z-axis)

1

.

1000

0100

00

00

1

'

'

'

z

y

x

CosSin

SinCos

z

y

x

PRP ).(' z

zz

yxy

yxx

'

cossin'

sincos'

x-axis rotation

P=(x,y,z)

x

y

z

3D Rotation(x-axis)

3D Rotation(x-axis)

PRP ).(' x

1

.

1000

00

00

0001

1

'

'

'

z

y

x

CosSin

SinCos

z

y

x

xx

zyz

zyy

xzyx

'

cossin'

sincos'

y-axis rotation

P=(x,y,z)

x

y

z

3D Rotation(y-axis)

3D Rotation(y-axis)

11000

00

0010

00

1

'

'

'

z

y

x

CosSin

SinCos

z

y

x

PRP ).(' y

yy

xzx

xzz

'

cossin'

sincos'

General 3D Rotation Rotation about an axis that is parallel to

x-axes.

-Translate object so that the rotation axis coincides with the parallel coordinate axis.

-Perform specified rotation about the axis.

-Translate the object so that the rotation axis is moved back to its original position.

PTRTP 1 .)(' x

Rotation about an axis that is parallel to x-axes.

Rotation axis

x

y

z

General 3D Rotation

Rotation about an axis that is not parallel to one of the coordinate axes.

-Translate object so that the rotation axis passes through the coordinate origin.

-Rotate object so that the axis of rotation coincides with one of the coordinate axes.

-Perform the specified rotation about that coordinate axis

-Apply inverse rotations to bring the rotation axis back to its original orientation.

-Apply the inverse translation to brig the rotation axis back to its original position.

Rotation about an axis that is not parallel to one of the coordinate axes.

Rot

atio

n ax

isx

y

z

P2

P1

Initial Position

Rotation about an axis that is not parallel to one of the coordinate axes.

x

y

z

P2

P1

Translate P1 to the origin

Rotation about an axis that is not parallel to one of the coordinate axes.

x

y

z

P 2

P 1

Rotate P2 onto the z-axis and rotate the object around it.

Rotation about an axis that is not parallel to one of the coordinate axes.

x

y

z

P2

P1

Rotate the axis to the original orientation

Rotation about an axis that is not parallel to one of the coordinate axes.

x

y

z

P2

P1

Translate the rotation axis to the original position

Rotation about an axis that is not parallel to one of the coordinate axes.

An axis vector is defined by two points as

A unit vector is defined as

1PPV 2

),,( cbaVV

u

Rotation about an axis that is not parallel to one of the coordinate axes.

The first step is to set up the translation matrix

1000

100

010

001

1

1

1

z

y

x

T

Rotation about an axis that is not parallel to one of the coordinate axes. The next step is to put the rotation axis

on the z-axis.

x

y

z

u

Rotation about an axis that is not parallel to one of the coordinate axes. First, we rotate about the x-axis to transform u into the x-z

plane.

x

y

z

u

α

Rotation about an axis that is not parallel to one of the coordinate axes.

First, we rotate about the x-axis to transform u into the x-z plane.

x

y

z

u

α

u′

Uz=(0,0,1)

α

22

'

')(

),,(

cbd

d

cCos

cba

z

z

uu

uu

u

Rotation about an axis that is not parallel to one of the coordinate axes. We can define the sine of α from the cross product of u and u′.

dbSin

dSinb

Sindb

dand

bcb

Sin

xx

z

x

zyx

z

zxz

/

)1)((

'1

100

0'

''

uu

uu

u

uuu

uu

uuuuu

Rotation about an axis that is not parallel to one of the coordinate axes.

The rotation matrix is

1000

0//0

0//0

0001

)(dcdb

dbdcx R

Rotation about an axis that is not parallel to one of the coordinate axes.

x

y

z

u

β

Rotation about an axis that is not parallel to one of the coordinate x-axes.

1000

00

0010

00

)(da

ad

y R

Rotation about an axis that is not parallel to one of the coordinate x-axes.

TRRRRRTR

R

)()()()()()(

1000

0100

00sin

00sin

)(

111

xyzyx

z

Cos

Cos