1Tranformations

Some slides adapted from Octavia Camps

2Matrices

nmnn

m

m

m

mn

aaa

aaa

aaa

aaa

A

21

33231

22221

11211

mnmnmn BAC Sum:Sum:

ijijij bac

A and B must have the same A and B must have the same dimensionsdimensions

3Matrices

pmmnpn BAC Product:Product:

m

kkjikij bac

1

A and B must have A and B must have compatible dimensionscompatible dimensions

nnnnnnnn ABBA

Identity Matrix:

AAIIAI

100

010001

4Matrices

Associative T*(U*(V*p)) = (T*U*V)*p Distributive T*(u+v) = T*v + T*v

5Matrices

mnT

nm AC Transpose:Transpose:

jiij ac TTT ABAB )(

TTT BABA )(

IfIf AAT A is symmetricA is symmetric

6MatricesDeterminant:Determinant: A must be squareA must be square

3231

222113

3331

232112

3332

232211

333231

232221

131211

detaa

aaa

aa

aaa

aa

aaa

aaa

aaa

aaa

122122112221

1211

2221

1211det aaaaaa

aa

aa

aa

7Matrices

IAAAA nnnnnnnn

11

Inverse:Inverse: A must be squareA must be square

1121

1222

12212211

1

2221

1211 1aa

aa

aaaaaa

aa

8Euclidean transformations

92D Translation

ttPP

PP !!

10

2D Translation Equation

PP

xxyy

ttxx

ttyyPP !!tt

tPP ),(' yx tytx

),(),(

yx tt

yx

tP

11

2D Translation using Matrices

PP

xxyy

ttxx

ttyyPP !!tt

),(),(

yx tt

yx

tP

110

01

' yx

t

t

tytx

y

x

y

xP

tt PP

12

Scaling

PP

PP !!

13

Scaling Equation

PP

xxyy

s.xs.x

PP !!s.ys.y

),('),(sysx

yx

PP

PP s'

yx

s

s

sysx

00

'P

SPSP '

14

RotationPP

PP !!

15

Rotation EquationsCounterCounter--clockwise rotation by an angle clockwise rotation by an angle

yx

yx

cossinsincos

'

'

PPxx

YY !!PP !!

XX !!yy R.PP'

16

Why does multiplying points by R rotate them? Think of the rows of R as a new coordinate system. Taking inner products of each points with these expresses that point in that coordinate system.

This means rows of R must be orthonormal vectors (orthogonal unit vectors).

Think of what happens to the points (1,0) and (0,1). They go to (cos theta, -sin theta), and (sin theta, cos theta). They remain orthonormal, and rotate clockwise by theta.

Any other point, (a,b) can be thought of as a(1,0) + b(0,1). R(a(1,0)+b(0,1) = Ra(1,0) + Ra(0,1) = aR(1,0) + bR(0,1). So its in the same position relative to the rotated coordinates that it was in before rotation relative to the x, y coordinates. That is, its rotated.

17

Degrees of Freedom

R is 2x2 R is 2x2 4 elements4 elementsBUT! There is only 1 degree of freedom: BUT! There is only 1 degree of freedom:

1)det(

RIRRRR TT

The 4 elements must satisfy the fol lowing constraints:The 4 elements must satisfy the fol lowing constraints:

yx

yx

cossinsincos

'

'

18

Transformations can be composed

Matrix multiplication is associative. Combine series of transformations into

one matrix. (example, whiteboard). In general, the order matters. (example,

whiteboard). 2D Rotations can be interchanged.

Why?

19

Rotation and Translation

100

tycos sin tx-sin cos ( )(

1yx )

100

tyabtx-ba( )(

1yx )

Rotation, Scaling and Translation

20

Rotation about an arbitrary point

Can translate to origin, rotate, translate back. (example, whiteboard).

This is also rotation with one translation. Intuitively, amount of rotation is same

either way. But a translation is added.

21

Inverse of a rotation If R is a rotation, RRT = I.

This is because the diagonals of RRT are the magnitudes of the rows, which are all 1, because the rows are unit vectors giving directions.

The off-diagonals are the inner product of orthogonal unit vectors, which are zero.

So the transpose of R is its inverse, a rotation of equal magnitude in the opposite direction.

22

Stretching Equation

PP

xxyy

SSxx.x.x

PP !!SSyy.y.y

yx

s

s

ysxs

y

x

y

x

00

'P

),('),(

ysxsyx

yx

PP

SPSP '

23

Stretching = tilting and projecting(with weak perspective)

yx

s

s

syx

s

s

ysxs

y

x

yy

x

y

x

10

00

0'P

24

Linear Transformation

yx

s

s

s

yx

s

s

yx

dcba

y

x

y

y

x

sincoscossin

10

0sincoscossin

sincoscossin

00

sincoscossin

'PSVD

25

Affine Transformation

1' y

x

tydctxba

P

26

Viewing Position Express world in new coordinate

system. If origins same, this is done by taking

inner product with new coordinates. Otherwise, we must translate.

27

Suppose, for example, we want to have the y axis show how we are facing. We want to be at (7,3), facing in direction (ct,st). The x axis must be orthogonal, (-st,ct). If we want to express (x,y) in this coordinate frame, we need to take: (ct,st)*(x-7,y-3), and (-st,ct)*(x-7,y-3). This is done by multiplying by matrix with rows (-st,ct) and (ct,st)

28

Simple 3D Rotation

n

n

n

zzz

yyyxxx

21

21

21...

1000cossin0sincos

Rotation about z axis.Rotates x,y coordinates. Leaves z coordinates fixed.

29

Full 3D Rotation

cossin0sincos0

001

cos0sin010

sin0cos

1000cossin0sincos

R

Any rotation can be expressed as combination of three rotations about three axes.

100010001

TRR Rows (and columns) of R are orthonormal vectors.

R has determinant 1 (not -1).

30

Intuitively, it makes sense that 3D rotations can be expressed as 3 separate rotations about fixed axes. Rotations have 3 degrees of freedom; two describe an axis of rotation, and one the amount. Rotations preserve the length of a vector, and the angle between two vectors. Therefore, (1,0,0), (0,1,0), (0,0,1) must be orthonormal after rotation. After rotation, they are the three columns of R. So these columns must be orthonormal vectors for R to be a rotation. Similarly, if they are orthonormal vectors (with determinant 1) R will have the effect of rotating (1,0,0), (0,1,0), (0,0,1). Same reasoning as 2D tells us all other points rotate too.

Note if R has determinant -1, then R is a rotation plus a reflection.

31

3D Rotation + Translation

Just like 2D case

32

Transformation of lines/normals

2D. Line is set of points (x,y) for which (x,y).(ab)T=0. Suppose we rotate points by R. We want a matrix, T, so that:

R*(x,y) .T

33

3D Viewing Position Rows of rotation matrix correspond to

new coordinate axis.

34

Rotation about a known axis

Suppose we want to rotate about u. Find R so that u will be the new z axis.

u is third row of R. Second row is anything orthogonal to u. Third row is cross-product of first two. Make sure matrix has determinant 1.