Geometric TransformationsHearn & Baker Chapter 5

Some slides are taken from Robert Thomsons notes.

OVERVIEW

• Two dimensional transformations• Matrix representations• Inverse transformations• Three dimensional transformations

Geometric transformation functions

• Translation• Rotation• Scaling• Reflection• Shearing

• We first discuss the 2D transformations, then we will continue with 3D.

2D Translations

TPP

d

dT

y

xP

y

xP

dyydxx

yxP

yxPP

yx

Now

,,

torscolumn vec theDefine

axis.y toparallel d axis, x toparallel d distance a ),(Point totranslate

),,( as defined Point

y

x

yx

PP’

2D Scaling from the origin.

y

x.

0

0

y

xor

Now

0

0

matrix theDefine

. ,.

axis.y thealong s and

axis, x thealong sfactor aby ),(Point to(stretch) scale a Perform

),,( as defined Point

y

x

y

x

y

x

yx

s

sPSP

s

sS

ysyxsx

yxP

yxPP

PP’

2D Rotation about the origin.

y

x

r

r

P’(x’,y’)

P(x,y)

2D Rotation about the origin.

y

x

r

r

P’(x’,y’)

P(x,y)

y

sin.

cos.

ry

rx

x

2D Rotation about the origin.

y

x

r

r

P’(x’,y’)

P(x,y)

y

sin.

cos.

ry

rx

x

cos.sin.sin.cos.)sin(.

sin.sin.cos.cos.)cos(.

rrry

rrrx

2D Rotation about the origin.

sin.

cos.

ry

rx

cos.sin.sin.cos.)sin(.

sin.sin.cos.cos.)cos(.

rrry

rrrx

Substituting for r :

Gives us :

cos.sin.

sin.cos.

yxy

yxx

2D Rotation about the origin.

cos.sin.

sin.cos.

yxy

yxx

Rewriting in matrix form gives us :

y

x

y

x.

cossin

sincos

PRPR

,

cossin

sincosmatrix theDefine

Transformations.

• Translation.– P=T + P

• Scale– P=S P

• Rotation– P=R P

• We would like all transformations to be multiplications so we can concatenate them express points in homogenous

coordinates.

Homogeneous coordinates

• Add an extra coordinate, W, to a point.– P(x,y,W).

• Two sets of homogeneous coordinates represent the same point if they are a multiple of each other.– (2,5,3) and (4,10,6) represent the same point.

• At least one component must be non-zero (0,0,0) is not defined.

• If W 0 , divide by it to get Cartesian coordinates of point (x/W,y/W,1).

• If W=0, point is said to be at infinity.

Homogeneous coordinates

• If we represent (x,y,W) in 3-space, all triples representing the same point describe a line passing through the origin.

• If we homogenize the point, we get a point of form (x,y,1)– homogenised points form a plane at W=1.

P

X

Y

W

W=1 plane

Translations in homogenised coordinates

• Transformation matrices for 2D translation are now 3x3.

1

.

100

10

01

1

y

x

d

d

y

x

y

x

11

y

x

dyy

dxx

Concatenation.

• We perform 2 translations on the same point:

),(),(),(

:expect weSo

),(),(),(

),(

),(

21212211

21212211

22

11

yyxxyxyx

yyxxyxyx

yx

yx

ddddTddTddT

ddddTPddTddTPP

ddTPP

ddTPP

Concatenation.

?

100

10

01

.

100

10

01

: is ),(),(product matrix The

2

2

1

1

2211

y

x

y

x

yxyx

d

d

d

d

ddTddT

Matrix product is variously referred to as compounding, concatenation, or composition

Concatenation.

Matrix product is variously referred to as compounding, concatenation, or composition.This single matrix is called the Coordinate Transformation Matrix or CTM.

100

10

01

100

10

01

.

100

10

01

: is ),(),(product matrix The

21

21

2

2

1

1

2211

yy

xx

y

x

y

x

yxyx

dd

dd

d

d

d

d

ddTddT

Properties of translations.

),(),(T 4.

),(),(),(),( 3.

),(),(),( 2.

)0,0( 1.

1-yxyx

yxyxyxyx

yyxxyxyx

ssTss

ssTttTttTssT

tstsTttTssT

IT

Note : 3. translation matrices are commutative.

Homogeneous form of scale.

y

xyx s

sssS

0

0),(

Recall the (x,y) form of Scale :

100

00

00

),( y

x

yx s

s

ssS

In homogeneous coordinates :

Concatenation of scales.

!multiply easy to -matrix in the elements diagonalOnly

100

0ss0

00ss

100

0s0

00s

.

100

0s0

00s

: is ),(),(product matrix The

y2y1

x2x1

y2

x2

y1

x1

2211

yxyx ssSssS

Reflection

corresponds to negative scale factors

originalsx = -1 sy = 1

sx = -1 sy = -1 sx = 1 sy = -1

Homogeneous form of rotation.

1

.

100

0cossin

0sincos

1

y

x

y

x

)()(

: i.e ,orthogonal are matricesRotation

).()(

matrices,rotation For

1

1

TRR

RR

i.e. the inverse is the transpose

Orthogonality of rotation matrices.

100

0cossin

0sincos

)( ,

100

0cossin

0sincos

)(

TRR

100

0cossin

0sincos

100

0cossin

0sincos

)(

R

Other properties of rotation.

matters.order Otherwise

same, thearerotation of centres theifonly

)()()()(

and

)()()(

)0(

RRRR

RRR

IR

2D Composite Transformations

• Combine transformations of different type– translate, rotate, translate– translate, scale, translate– translate, reflect, translate

• Use to rotate or scale an object w.r.t. a point that is not the origin

• Implement by multiplication of the corresponding homogeneous matrices

reflection in x and y axes reflection in origin

1 0

1

x x

y xb b y

1

0 1

x x y

y y

a a

Shx gives the slope of a vertical line after shear, as dx/dy

Example 1

y

yx

x

y

xyxT

4.0

14.0

01),(

)1,0(

)0,1(

)1,1(

)0,0(

S

x

Example 1

x

y

yx

x

yxT

4.0

),(

S

)1,0(

)0,1(

)1,1(

)0,0(

Example 1

y

)1,0(

)4.0,1(

)0,0(

T(S)

yx

x

yxT

4.0

),()4.1,1(

x

Example 2

0.6

0.6

1( , )

0 1

xT x y

y

x y

y

y

Sx

)1,0(

)0,1(

)1,1(

)0,0(

Example 2

0.6( , )

x yT x y

y

x

y

S

)1,0(

)0,1(

)1,1(

)0,0(

Example 2

( , ) 0.6x y

T x yy

x

y

)1,0(

)0,1()0,0(

T(S)

(0.6,1) (1.6,1)

Summary

Shear in x:

Shear in y:

y

yshx

y

xshSh xx

x 10

1

yxsh

x

y

x

shSh

yyy .1

01

Double Shear: not commutative!

ab)(10

1

1

01

1

1

b

aa

b

1

1 ab)(

1

01

10

1

b

a

b

a

Shear Matrix defined by angle, not slope

e.g. simple shear along x axis

x’ = x + y cot y’ = yz’ = z

1000

0100

0010

00cot 1

H() =

3D Transformations.

• Use homogeneous coordinates, just as in 2D case.• Transformations are now 4x4 matrices.• We will use a right-handed (world) coordinate system -

( z out of page ).

z (out of page)

y

x

Note:Convenient to think of display asBeing left-handed !!( z into the screen )

Simple extension to the 3D case:

Scale in 3D.

1000

000

000

000

),,(z

y

x

zyx s

s

s

sssS

Simple extension to the 3D case:

Rotation in 3D

• Need to specify which axis the rotation is about.• z-axis rotation is the same as the 2D case.

1000

0100

00cossin

00sincos

)(

zR

Rotation around an axis parallel to x-axis

H&B p271

Rows of upper-left 3x3 submatrix, when rotated by R lie on the x,y and z axes

1

1

0

0

1

1

0

1

0

1

1

0

0

1

1

33

32

31

23

22

21

13

12

11

r

r

r

Rr

r

r

Rr

r

r

R

Scaling an object with this transformation will also move its position relative to the origin - so move it to the origin, scale it, then move it back...