1

Transformation

2

Matrix

column and row

•

identity matrix

1st row

2nd row

3rd row

1st

column

2nd

column

3rd column

transpose

matrix composition

M-1 is an inverse matrix of M if MM-1=I.

3x3

333231

232221

131211

aaa

aaa

aaa

M

332313

322212

312111

aaa

aaa

aaa

M T

232221

131211

3231

2221

1211

bbb

bbb

aa

aa

aa

233213312232123121321131

232213212222122121221121

231213112212121121121111

babababababa

babababababa

babababababa

100

010

001

I

3x2 2x3

3

Translation

dyy

dxx

dy

dx

y

x

y

x

(x, y)

( x, y)

dx

dy dyyy

dxxx

A transformation is a function that maps a point/vector into another point/vector.

For now, we will discuss the basic 4 types of transformations: translation, scaling, rotation, and shearing.

Translation

4

Scaling

y

x

s

s

y

x

y

x

0

0

e.g.

x

y

(6,2)Scale by 1/2 along x

Scale by 2 along y

3 6

(3,4)4

Scaling is done about the origin. Scale x by sx along x-axis: x = sx·x Scale y by sy along y-axis: y = sy·y

The above example is called a uniform scaling where sx = sy. If sx sy, non-uniform scaling.

If 0 < sx (or sy) < 1, objects are moved closer to the origin. If sx or sy > 1, objects are moved farther from the origin.

e.g.

x

y

(6,4)

3 6x 2

4(3,2)

2x 2

5

Reflection Reflection or mirror matrix: a scaling matrix where (sx,sy) is either

(1,-1) or (–1,1). (If both sx and sy are –1, it’s a 180o rotation.)

y

x

y

x

10

01

y

x

y

x

10

01x

y

y

x

y

x

10

01

x

y

x

y

x

y

6

Rotation

Rotation by an angle about the origin.

The distance from P to the origin is r. Angles are measured CCW. x = r cos y = r sin x = r cos(+) = r cos cos - r sin sin = xcos - ysin y = r sin(+) = r cos sin + r sin cos = xsin + ycos For CW, you may use cos(-)=cos and sin(-)=-sin.

y

x

y

x

cossin

sincos

P(x, y)

P (x, y)

r

rr

7

Shearing

Shearing along x

y = y (unaffected)x = x + ay (in the example, a 0.5)

y

xa

y

x

10

1

Similarly, shearing along y

•

y

x

by

x

1

01

x

y

x

y

8

Homogeneous Coordinates

Observe that rotation, scaling & shearing (which are collectively called linear transformations) are done by “matrix composition” while translation is by “vector addition.” Can we treat them uniformly?

YES!! Use Homogeneous Coordinates.

Practical motivations for Homogeneous Coordinates. We can treat {rotation, scaling, shearing, etc.} & {translation} uniformly.

We can deal with projections (later!).

A point (x, y) in Cartesian Coordinates is represented as (wx, wy, w) in Homogeneous Coordinates where w can be any non-zero value.

e.g. (2 3) (2 3 1) = (4 6 2) = (5 7.5 2.5) = …... but, typically w=1.

9

Transformations in Homogeneous Coordinates

translation (x = x+dx, y = y+dy) scaling

rotation shearing (along x)

11100

10

01

1

dyy

dxx

y

x

dy

dx

y

x

11100

00

00

1

ys

xs

y

x

s

s

y

x

y

x

y

x

1100

0cossin

0sincos

1

y

x

y

x

1100

010

01

1

y

xa

y

x

Note that all transformations are done by “matrix composition” thanks to homogeneous coordinates.

Transformations which do not alter a body’s shape are called rigid-body transformations, and consist of rotations & translations.

10

Rotation about An Arbitrary Point

How to rotate about an arbitrary point p(x, y), not about the origin?

translatetranslate & rotatetranslate, rotate & back-translate

translate q by -p rotate by back-translate

p =(x,y)

q=(a,b)

q=(a,b)

p

q

p=O

q

p=O

qq q

q

3x3 3x3 3x3

11100

10

01

100

0cossin

0sincos

100

10

01

b

a

b

a

y

x

y

x

Thanks to Homogeneous Coordinates, three transformations can be combined into a single 3x3 matrix.

11

Inverse Transformation Note that the first and last translations are inverses of each other.

Such an inverse transformation is represented as an inverse matrix, but can be obtained pretty easily if its geometry is considered.

translate q by -p rotate by back-translate

p

q

p=O

q

p=O

qq q

q

inverse transformations

100

10

01

100

10

011

y

x

y

x

d

d

d

d

100

10

01

100

10

011

y

x

y

xx'=x+dx

x'+?=x?=- dx

12

100

010

01

100

010

011

aa

Inverse Transformation (cont’d)

scaling

rotation

shearing (along x) We have to restore y'(=y) into y and x'(=x+ay) into x, respectively. x'' = x' + ?y' = x' + ?y = x + ay + ?y = x. So, ?=-a

x'=xxsx

x'x?=x?=1/sx

100

01

0

001

100

00

001

y

x

y

x

s

s

s

s

100

0cossin

0sincos

100

0)cos()sin(

0)sin()cos(

100

0cossin

0sincos1

x

y

13

Associativity & Commutativity

Matrix multiplication is associative. So is transformation composition.

ABC = (AB)C = A(BC)

In contrast, matrix multiplication is not commutative in general. Neither is transformation composition.

• e.g. combination of rotation (by 90°CCW) & translation along y

• rotation first translation first

TRpp RTpp pp

14

3D Translation and Scaling

First of all, Homogeneous Coordinate for 3D (x y z) → (wx wy wz w) e.g (x y z 1)

Translation

Scaling

111000

100

010

001

dzz

dyy

dxx

z

y

x

dz

dy

dx

111000

000

000

000

zs

ys

xs

z

y

x

s

s

s

z

y

x

z

y

x

dzzz

dyyy

dxxx

zsz

ysy

xsx

z

y

x

15

3D Reflection

Reflection or mirror matrix: a scaling matrix where one scaling factor is –1 and two others are 1 or all of the three scaling factors are –1.

If two are –1, it’s a 180o rotation.

1000

0100

0010

0001

1000

0100

0010

0001

1000

0100

0010

0001

1000

0100

0010

0001

reflection wrt yz-plane

reflection wrt xz-plane

reflection wrt xy-plane

reflection wrt the origin

16

3D Rotation For 3D rotation, we need an axis of rotation. Also, by convention, positive angles are for CCW. Recall that, in 2D, x = xcos - ysin and y = xsin + ycos. We can consider (x,y) and (x,y) as points at xy-plane of 3D space.

•

Add z-axis and make z = z. Then, it’s 3D rotation about the z-axis.

y

x

y

x

z Rz(θ)

(x, y)(x, y)

(x, y, z)(x, y, z)

111000

0100

00cossin

00sincos

z

y

x

z

y

x

y

x

z

(x, y, 0)(x, y, 0)

17

zz

yxy

yxx

cossin

sincos

3D Rotation (cont’d) Rotation about x-axis

Note that the coordinate system follows the right hand rule. Let’s do cyclic permutation of the 3 axes, where the right hand rule holds.

Note that rotation about x-axis is the same as rotation about the old z-axis except the permutation .

For rotation about y-axis, do one more permutation! 3D shearing will be discussed when we deal with viewing.

111000

0cossin0

0sincos0

0001

z

y

x

z

y

x

yz

zx

xy

y' y z

yz' zx' x

xx

zyz

zyy

cossin

sincos

xyzx

18

Geometric Transformation of a Mesh

Apply transformations to every vertex. Geometry is transformed. Topology(connectivities) is not changed.

An example, Mesh A is displayed by drawing all of its edges (wireframe display). All vertices of A are rotated, scaled up and translated to make B, and

then wireframe-displayed.

A B

19

3D Rotation about An Arbitrary Axis

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

Step1: Translate such that the rotation axis passes

through the origin.

Step2: Rotate such that the rotation axis coincides

with a coordinate axis.

Step4: Inverse rotation. (Inverse of Step2)

Step5: Inverse translation. (Inverse of Step1)

Step3: Rotate by

by

20

3D Rotation about An Arbitrary Axis (cont’d)

Rotation axis is typically defined by 2 end points p1 and p2 (or a point p and a vector v) Move ∴ p1=(x1 y1 z1) to the origin.

Consider the unit vector u along the rotation axis.

1000

100

010

001

1

1

1

z

y

x

T

x

y

zp1(x1 y1 z1)

p2(x2 y2 z2)

p2 (x2 y2 z2)

p1 (x1 y1 z1)

x

y

z

unit vector u x

y

z

p1(x1 y1 z1)

p2(x2 y2 z2)

p2 (x2 y2 z2)

p1 (x1 y1 z1) z

x

y

)( 121212

12

zzyyxx

ppU

)( cbaU

Uu

21

Step2 (from u to uz) requires a rather complicated process, which will be discussed soon. Just suppose the rotation is denoted by R.

Step3 is simply

Step4: an inverse of the matrix computed at Step2 Step5: an inverse of the matrix computed at Step1 R() is a single 4x4 matrix.

3D Rotation about An Arbitrary Axis (cont’d)

1000

0100

00cossin

00sincos

)(

zR

TRRRTR z )()( 11

y

x

z

y

x

z

Ru

uz

22

Orthogonal Matrix Note that rotations we’ve devised are all about coordinate axes. Rotation for Step 2 is presented though orthogonal matrix.

2 rows vi & vj are orthonormal if i.e. each row is a unit vector, and ①• ② any two different rows are orthogonal.

Orthogonal matrix = a matrix with orthonormal rows.

• e.g.

1...000

:::::

0...100

0...010

0...001

nn

100

02/12/1

02/12/1

33

)(1

)(0

ji

ji

ji vv

y

x

z

y

x

z

Ru

uz

23

Orthogonal Matrix (cont’d) A matrix Q is orthogonal Q·QT = I (identity matrix)

Observations An orthogonal matrix transforms its rows into coordinate axes. QT = Q-1, i.e. for orthogonal matrices, the transpose is the inverse. It’s

great because inverse matrix computation is usually expensive. if Q is an orthogonal matrix, QT is also an orthogonal matrix.

Q-1·Q = I = QT·Q QT·(QT)T = I QT is orthogonal.

1...

0...010

0...001

......

nq

1q

2q0...100... ... ... ...

0 0 0

T1q T

2q Tnq

nxn nxn nxn nxn nx1 nx1

...

nq

1q

2q T1q

0

1

0...

0

e.g.

24

3D Orthogonal Matrix The same thing applies to 3D. Suppose that we have 3 orthonormal vectors as follows.••

We can make orthogonal matrices using the 3 orthonormal vectors. If the 3 vectors constitute a coordinate system, i.e. the cross product

of the 1st row and the 2nd row corresponds to the 3rd row following the right-hand rule, the orthogonal matrix is a rotation.

ux

uy

uz

v3

v2

v1

v1

v2

v3

v3

v1

v2

v1

v3

v2

25

3D Orthogonal Matrix (cont’d)

0

0

0

1

013

12

11

r

r

r

R

v1 ux

0

0

1

0

023

22

21

r

r

r

R

v2 uy

0

1

0

0

033

32

31

r

r

r

R

v3 uz

3D rotation in homogeneous coordinates.•

•

R transforms v1, v2 and v3 into ux, uy and uz, respectively.

1000

0

0

0

333231

232221

131211

rrr

rrr

rrr

R

v1

v2

v3 ux

uy

uz

v3

v2

v1

‘an’ orthogonal matrix

26

3D Rotation about An Arbitrary Axis (cont’d)

z

1000

0

0

0

3

2

1

u=v

v

v

u =

0

1

0

0

0

zu

u = v3

y

xxu

yu

zu

Recall that, after Step1, our goal was to rotate u into uz.

• For this purpose, let’s make a rotation matrix whose 3rd

row is u.

27

3D Rotation about An Arbitrary Axis (cont’d)

Let’s simply put R into R()=T-1R-1Rz()RT. Importantly, RT is used for R-1.

321

2

3

vvv

uu

uuv

uv

x

x

u = v3

uz

ux

uy

y

x

z v1

v2

is used to make be perpendicular to .

You can use any coordinate axis vector if it is not equal to u .

xu 2v

3v

1000

0

0

0

3

2

1

u=v

v

v

R =

28

Modeling Transformation

29

Modeling Transformation

30

Modeling Transformation

A r p B

),(),('' yxTyxScaleM xvyx

Modeling Transformation

'y

'xx

p B A r

y

x

y

31

Modeling Transformation

p B A

y

xp B A r

y

x

Translate by -r, bringing r to the origin.

p B A

y

x p B A

y

x

)(ARotate )(Atranslate

32

Modeling Transformation

translate both A and B by –p,bringing p to the origin.

p B A

y

xp B A

y

x

Rotate A&B Translate

p B A

y

x

p B

A

y

x

33

Modeling Transformation