Transformations - Technion – Israel Institute of...

TransformationsTransformations

Geometric Transformations

� Transformation τ is one-to-one and onto mapping of

Rn to itself (in general)

� Affine transformation – τ (V) = AV+b

A – matrix

b – scalar

Transformations Transformations -- Center for Graphics and Geometric Computing, Center for Graphics and Geometric Computing,

TechnionTechnion

3

b – scalar

� In CG Geometric Transformations are, for the most

part, affine transformations with geometric meaning

� Note: transformation has meaning only for vectors ⇒for point P, τ (P) is a transformation of vector from the

origin to P

Transformations

� Transforming an object = transforming all its points

� For a polygonal model = transforming its vertices

Is it true for any transformation?

Why is it true for us?


TechnionTechnion

4

Applications

� Viewing

� Modeling

� Animation


TechnionTechnion

5

� Animation

� Image manipulation

Scaling

� V = (vx,vy) – vector in XY plane

� Scaling operator S with parameters (sx,sy):

( )( ) ( )yyxx

sssvsvVS yx ,

, =


TechnionTechnion

6

( ) ( )yyxx svsvVS ,=

Scaling� Matrix form:

( )( ) ( ) ( )yyxx

y

x

yx

sssvsv

s

svvVS yx ,

0

0,

, =

=


TechnionTechnion

7

� Independent in x and y

Non uniform scaling is allowed

� What is the meaning of scaling by zero?

Rotation

� Polar form:

W

y

V

x

r

αθ


TechnionTechnion

8

� Polar form:

V=(vx,vy)=(r cosα, r sinα)

� Rotating V anti-clockwise by θ to W:

)cossinsincos

,sinsincoscos(

))sin(),cos((

),(

θαθαθαθα

θαθα

rr

rr

rr

yw

xwW

+−=

++=

=

Rotation

� Matrix form:

−=

−=

θθθθ

θθθθ

αα

cossin

sincos

cossin

sincos)sin,cos(

V

rrW

)cossinsincos

,sinsincoscos(

θαθαθαθα

rr

rrW

+−=


TechnionTechnion

9

� Rotation operator R with parameter θ at the origin:

−=

θθθθ

θ

cossin

sincosR

Rotation Properties

� Rθ is orthogonal

� R-θ - rotation by -θ is

( ) ( )TRR θθ =−1


TechnionTechnion

10

� R - rotation by -θ is

( ) 1

cossin

sincos),()(

−− =

−= θθ

θθθθ

RvvVR yx

Translation


TechnionTechnion

11

� Translation operator T with parameters (tx,ty):

� Can you express T in a matrix form?

( )( ) ( )yyxx

tttvtvVT yx ++= ,

,

Translation - Homogeneous

Coordinates� To represent T in matrix form

– introduce homogeneous

coordinates:

� Conversion (projection) from

( ) ( )1,,,, h

y

h

x

h

w

h

y

h

x

h vvvvvV ==


TechnionTechnion

12

� Conversion (projection) from

homogeneous space to

Euclidean:

� In homogeneous

coordinates:

(2, 2, 1) = (4, 4, 2) = (1, 1, 0.5)

==

h

w

h

y

h

w

h

xyx

v

v

v

vvvV ,),(

Translation

� Using homogeneous coordinates, the translation

operator may be expressed as:

001


TechnionTechnion

13

)1,,(

1

010

001

)1,,()(),(

yyxx

yx

yx

htt

tvtv

tt

vvVTyx

++=

=

Matrix Form

�Why did we insist on using a Matrix form to

represent the transformations?

Hint: How would you go about transforming

an object of m vertices through a sequence of


TechnionTechnion

14

an object of m vertices through a sequence of

n transformations?

Transformation Composition

� What operation rotates XY by θ around P = (px,py) ?

� Answer:

Translate P to origin

Rotate around origin by θ


TechnionTechnion

15

Rotate around origin by θ

Translate back

Transformation Composition

=−−

0010sincos001

),(),( ppppTRT yxyx

θθθθ

θ


TechnionTechnion

16

−− 1

010

100

0cossin-

1

010

yxyx pppp

θθ

Transformations Quiz

� What do these

transformations do?

� And these homogeneous

ones?

−

01

10

01

5.000

010

001

1-00

010

001


TechnionTechnion

17

� How can one mirror a planar

object through an arbitrary

line in XY?

� Can one rotate a planar

object in the XY plane via

mirroring?

−

−

10

1

11

01

10

01

10

a

Rotate by Shear

� Shear

� Rotation by 0 ≤ θ ≤ π/2 ≡ composition of 3 shears

1011sincos zxθθ

1

01,

10

1,

10

11,

11

01

a

a


TechnionTechnion

18

.1

1

10

1

1

1

10

1

1

01

10

1

cossin-

sincos

+

+++=

+=

=

yzy

xxyzzxy

z

y

xxy

z

y

x

θθθθ

Rotate by Shear (cont.)

� Solve for x, y, z :

cos θ = 1+xy = yz+1

sin θ = −y = z+xyz+x

.1

1

cossin-

sincos

+

+++=

yzy

xxyzzxy

θθθθ


TechnionTechnion

19

� Solution:

y = −sin θx = z = −tan(θ / 2)

QuestionQuestion: When is rotation-using-three-shears useful?

QuestionQuestion: Can we derive a rotation via two shears?

Rotate by Shear (cont.)

� What does the composition of

the following two scaled shears

do?

=

θθθθ sincos0secsin1


TechnionTechnion

20

� Why use it?

� Is it convenient?

� What happens when θ → π/2 ?

=

−

θθθθ

θθ

θθ

cossin-

sincos

1tan

0sec

cos0

sin1

Rotation Approximation

� For small angles can approximate:

x’=x cosθ − y sinθ ≅ x − yθy’=x sinθ + y cosθ ≅ xθ + y

Since cosθ → 1 and sinθ → θ as θ → 0 (Taylor expansion of sin/cos)

� Example (steps of θ ):


TechnionTechnion

21

� When is this rotation approximation useful?

� App. rotation matrix

� Matrix has to be orthogonal & normalized

Rotation Approximation

−

=1

1

θθθAR


TechnionTechnion

22

� Matrix has to be orthogonal & normalized

� ARθ is orthogonal

� ARθ not normalized

�Normalize

2

1

1

1 θθ

θθ +=

−= cAR

cc

cc

3D Transformations

� All 2D transformations extend to 3D

� In homogeneous coordinates:

Scale Translate Rotate (around z axis)


TechnionTechnion

23

�� QuestionQuestion: What is ? ?

−

=

=

=

1000

0100

00cossin

00sincos

1

0100

0010

0001

1000

000

000

000

),,(),,( θθθθ

θz

zyx

ttt

z

y

x

sssR

ttt

Ts

s

s

S zyxzyx

θxR

θyR transformations

3D Transformations

� Questions (commutativity):

Scaling: Is S1S2 = S2S1?

Translation: Is T1T2 = T2T1?


TechnionTechnion

24

Rotation: Is R1R2 = R2R1?

Example: Arbitrary Rotation

YZ

W

V


TechnionTechnion

25

� Problem:

Given two orthonormal coordinate systems XYZ and UVW ,

Find a transformation from one to the other.

X

U

Arbitrary Rotation

� Answer: Transformation matrix R

whose rows are U, V, W:

� Proof:

uuu

=

zyx

zyx

zyx

www

vvv

uuu

R


TechnionTechnion

26

� Similarly R(Y) = V & R(Z) = W

U

uuu

www

vvv

uuu

XR

zyx

zyx

zyx

zyx

=

=

=

),,(

)0,0,1()(

� Inverse (=transpose) transformation, R-1, provides

mapping from UVW to XYZ

� E.g.

Arbitrary Rotation (cont.)

wvu

wvu

uuuUR yyy

xxx

zyx

=− ),,()(1


TechnionTechnion

27

� Comment: Used for mapping between XY plane and

arbitrary plane (why is this useful!?)

X

uuu

wvu

wvuuuuUR

zyx

zzz

yyyzyx

=

=

++=

=

)0,0,1(

)0,0,(

),,()(

222

� Lemma: Affine image of line segment is line segment

between image of its end points

� Proof:

Given affine transformation A & line segment S(t) = P1t + P2(1-t)

Transformation of Objects

))1(())(( 21 tPtPAtSA −+=


TechnionTechnion

28

� Is an affine image of a plane, a plane?

� Conclusion: polyhedron (polygon) is affinely transformed

by transforming its vertices

)()1()(

))1(()(

))1(())((

21

21

21

PAtPtA

tPAtPA

tPtPAtSA

−+=

−+=

−+=

Viewing Transformations� Question: How can we view (draw)

3D objects on a 2D screen?

� Answer:

Project the transformed object along Z

axis onto XY plane - and from there to

screen (space)

Canonical projection:ObjectObjectspacespace

ViewViewspacespace

ScreenScreenspacespace


TechnionTechnion

29

Canonical projection:

1000

0000

0010

0001

� In practice “ignore” the z axis – simply use the x and y

coordinates for screen coordinates

z

ObjectObjectspacespace

ScreenScreenspacespace

Parallel Projection


TechnionTechnion

30

Parallel Projection

� Projectors are all parallel

Orthographic: Projectors perpendicular to projection plane

Axonometric: Rotation + translation + orthographic projection


TechnionTechnion

31

Parallel Projection� Projectors are all parallel

Oblique projection: Projectors not necessarily perpendicular to

projection plane

transformations


TechnionTechnion

32

� When can the oblique projection used?

What is Wrong in this Korean Picture?


TechnionTechnion

33

Perspective Projection

� Viewing is from point at a

finite distance

� Without loss of generality:

Viewpoint at origin

projection

plane

X

p2 p3


TechnionTechnion

34

Viewing plane is z=d

� Given P=(x, y, z), triangle

similarity gives:

dz

yy

dz

xx

d

y

z

y

d

x

z

xpp

pp

/ and

/ and ==⇒==

center of

projection

projectors

x2p

x1p= x3

p

z = d

p1= (x1, y1, z1)

Z

� In matrix notation with homogeneous coordinates:

Perspective Projection (cont’d)

.,,,

0000

/1100

0010

0001

)1,,,()1,,,(

=

=d

zzyx

dzyxzyxP


TechnionTechnion

35

� In Euclidean coordinates:

� P not injective & singular: det(P)=0

0000

).,,(,/

,//

,/

,/

dyxddz

y

dz

x

dz

z

dz

y

dz

xpp=

=

transformations

Perspective Projection (cont’d)� What is (if any) the difference between:

Moving the projection plane

camtrans


TechnionTechnion

36

Moving the viewpoint (center of projection)?

Examples of Perspective projection

Perspective could be defined with

one, two or three focal points:


TechnionTechnion

37


The same model in

Orthographic view Perspective view


TechnionTechnion

38


The same model with

Small perspective warp Large perspective warp


TechnionTechnion

39

More than Three Points’ Perspective!?


TechnionTechnion

40

Perspective Warp� P not invertible – can not get depth (order) back.

� Idea:

Warp the viewing frustum (and the entire world)

Then use parallel projection


TechnionTechnion

41

Question: Why

do we require

the depth

information?

Perspective Warp

� Matrix formulation

−

−=

− d

z

d

dzyx

dd

dzyx ,

)(,,

100

0010

0001

)1,,,(α

α

α


TechnionTechnion

42

� This matrix maps the x and y coordinates much like

the perspective map in slide 34.

−

−−

− dd

d

ddd

000

α

αα

α

( )( )

−−

=zd

zd

dz

y

dz

xzyx ppp α

α2

,/

,/

),,(

Perspective Warp

� What does α = 0 mean? α = d ?

� What is the effect of changing d, d >α ?

� The perspective warp preserves relative depth (third

coordinate). For what depth domain? Prove it!

( )( )

−−

=zd

zd

dz

y

dz

xzyx ppp α

α2

,/

,/

),,(

camtrans


TechnionTechnion

43

coordinate). For what depth domain? Prove it!

� The perspective warp preserves planes (prove it!)

� Plane Ax + By + Cz + D = 0 is warped to (prove it!):

Aαdxp + Bαdyp + D(α − d)zp + Cαd2 + Dd2 = 0

� What does warp do to other objects? Spheres?

Perspective Warp� We will show that the six faces of the

view frustum are mapped to six faces

of a box

� Plane z = d (d ≠ 0), A=B=0, C=1, D=-d and the plane is

warped to -d(α -d)zp+α d2 - d3 = 0 or zp = d

Ax + By + Cz + D = 0 ⇒ Aαdxp + Bαdyp + D(α − d)zp + Cαd2 + Dd2 = 0


TechnionTechnion

44

p p

� Plane z = α (α ≠ 0), A=B=0, C=1, D=-α and the plane is

warped to -α(α -d)zp+α d2 -α d2 = 0 or zp = 0

� Plane x = ±z, A=1, B=0, C= ±1, D=0 and the plane is

warped to αdxp ± α d2 = 0 or xp = ± d

� Plane y = ±z, A=0, B=1, C= ±1, D=0 and the plane is

warped to αdyp ± α d2 = 0 or yp = ± d

What is the difference between

O = M No & O = N M o, N new transform

Another Transformations Quiz� What does each transformation preserve?

linesparallel

linesdistance angles normals convexity

Conics

(spherity)

scaling

rotation


TechnionTechnion

46

� A map that preserves angles is called conformal.

� A map that preserves distances is called an Isometry.

translation

shear

perspective

Everything is a matter of Perspective

Sidewalk Chalk PaintingsSidewalk Chalk Paintings


TechnionTechnion

47

Julian Julian BeeverBeever

Transformations - Technion – Israel Institute of...

Documents

Transcript of Transformations - Technion – Israel Institute of...