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Copyright 2004, A. Sheffer, UBC

Transformations

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Chapter 3

Transformations

transformations

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Geometric Transformations

Transformation τ is one-to-oneand onto mapping of nD to itself

Affine transformation – τ(V) = AV+bA – matrixb – scalar

In CG Geometric Transformations are affine transformations with geometric meaningNote: transformation has meaning only for vectors ⇒ for point P τ(P) is transformation of vector from (0,0) to P

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Transformations

Transforming an object = transforming all its points

Transforming a polygon = transforming its vertices

Why?

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Copyright 2004, A. Sheffer, UBC

Transformations

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Applications

Viewing

Modeling

Articulation

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Scaling

V = (vx,vy) – vector in XY plane

Scaling operator S with parameters (sx,sy):S(sx,sy)(V) = (vxsx, vysy)

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Scaling

Matrix form:

Independent in x and y

S(sx,sy)(V) = (vx, vy)sx 00 sy

ò ó= (vxsx, vysy)

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Copyright 2004, A. Sheffer, UBC

Transformations

University ofUniversity ofBritish ColumbiaBritish Columbia

Rotation

Polar form:

Rotating V counterclockwise by θ to W:

)cossinsincos,sinsincoscos(

))sin(),cos((

),(

θαθαθαθα

θαθα

rrrr

rrywxwW

+−=

++=

=

WyV

x

θ α

r

V = (vx, vy) = (rcosë, r sinë)

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Rotation

Matrix form:

Rotation operator R with parameter θ at the origin:

−=

θθθθ

θcossinsincos

R

W r r

V

=−

=−

( cos , sin )cos sinsin cos

cos sinsin cos

α αθ θθ θ

θ θθ θ

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Rotation Properties

Rθ is orthogonal

R-θ - rotation by -θ is

( ) ( )TRR θθ =−1

( ) 1

cossinsincos

),()( −− =

−= θθ

θθθθ

RvvVR yx

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Computer GraphicsComputer Graphics

Copyright 2004, A. Sheffer, UBC

Transformations

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Translation

Translation operator T with parameters (tx,ty):

T(tx,ty)(V) = (vx+ tx, vy+ ty)

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To represent T in matrix form – introduce homogeneous coordinates:

Conversion (projection) from homogeneous space to Euclidean:

In homogeneous coordinates:

Translation - Homogeneous Coordinates

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

V v v v v vhxh

yh

wh

x y= =( , , ) ( , , )1

V v vvv

vvx y

xh

wh

yh

wh= =

( , ) ,

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Translation

Using homogeneous coordinates, translationoperator may be expressed as:

( , ) ( ) ( , , )

( , , )

t t hx y

x y

x x y y

x yT V v vt t

v t v t

=

= + +

11 0 00 1 0

1

1

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Copyright 2004, A. Sheffer, UBC

Transformations

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Transformation Composition

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

Translate P to originRotate around origin by θTranslate back

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Transformation Composition

T R T

p p p p

p p p p

x y x y

x y x y( , ) ( , )

cos sin-sin cos

− − =

− −

θ

θ θθ θ

1 0 00 1 0

1

00

0 0 1

1 0 00 1 0

1

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Transformations Quiz

What do these transformations do?

And these homogeneous ones?

How to mirror through arbitrary line in XY?

What transformation achieves this?

−

−

101

1101

1001

1001

1001

a

5.000010001

1-00010001

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Computer GraphicsComputer Graphics

Copyright 2004, A. Sheffer, UBC

Transformations

University ofUniversity ofBritish ColumbiaBritish Columbia

Rotate by Shear

Shear

Rotation by composition of 3 shears

.1

1

101

11

101

101

101

cossin-sincos

++++

=

+=

=

yzyxxyzzxy

zy

xxy

zy

xθθθθ

0 ô ò ô 2ù ñ

101

,10

1,

1011

,1101

aa

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Rotate by Shear (cont.)

Solve for x, y,z :

Result

cosò =1+xy= yz+1, sinò =ày= z+xyz+x

y =à sinò,x = z =à tan2ò

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What does composition of following two scaled shears do?

Why use it?Is it convenient?What happens when ?

Rotate by Shear (cont.)

10

01

sincos

sectan

cos sinsin cos

θθ

θθ

θ θθ θ

−

=

−

2θ π→

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Computer GraphicsComputer Graphics

Copyright 2004, A. Sheffer, UBC

Transformations

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Rotation Approximation

For small angles can approximate:

as(Taylor expansion of sin/cos)

Example:

y0=x sin ò+y cos ò'xò+yx0=x cos òày sin ò'xàyò

cosò→ 1, sin ò→ ò ò→ 0

25° 10° 2° 0.5°

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Rotation Approximation

App. rotation matrix

Matrix has to be orthogonal & normalizedARθ is orthogonalARθ not normalizedNormalize

ARò = c1

cò

càò

c1

!, c = (1+ ò2)

p

ARò =1 òà ò 1

ò ó

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3D Transformations

All 2D transformations extend to 3D In homogeneous coordinates:

−

=

=

=

1000010000cossin00sincos

1010000100001

1000000000000

axis z thearoundRotation n Translatio Scaling

),,(),,( θθθθ

θz

zyx

ttt

z

y

x

sss R

ttt

Ts

ss

S zyxzyx

transformations

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Copyright 2004, A. Sheffer, UBC

Transformations

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3D Transformations

Questions:Is S1S2 = S2S1?

Is T1T2 = T2T1?

Is R1R2 = R2R1?

Is S1R2 = R2S1?

…..

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Example: Arbitrary Rotation

Problem: Given two orthonormal coordinate systems XYZand UVW Find transformation from one to the other

Answer: Transformation matrix R whose rows are U,V,W:

=

zyx

zyx

zyx

wwwvvvuuu

R

Y Z

X

W

V

U

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Arbitrary Rotation

Proof:

Similarly R(Y) = V & R(Z) = W

R Xu u uv v vw w w

u u u

U

x y z

x y z

x y z

x y z

( ) ( , , )

( , , )

=

=

=

1 0 0

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Copyright 2004, A. Sheffer, UBC

Transformations

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Inverse (=transpose) transformation R-1

provides mapping from UVW to XYZE.g.

Comment: Used for mapping between XY and arbitrary plane

Arbitrary Rotation (cont.)

X

uuu

wvuwvuwvu

uuuUR

zyx

zzz

yyy

xxx

zyx

==

++=

=−

)0,0,1(

)0,0,(

),,()(

222

1

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Lemma: Affine image of line segment is line segment between image of its end pointsProof:

Given affine transformation A & line segment

Conclusion: polyhedron(polygon) is affinelytransformed by transforming its vertices (Why?)

Transformation of Objects

)1()( 21 tPtPtS −+=

)()1()())1(()(

))1(())((

21

21

21

PAtPtAtPAtPA

tPtPAtSA

−+=−+=

−+=

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Viewing Transformations

Question: How to view (draw) 3D object on 2D screen?Answer:

Project transformed object along Z axis onto XY plane -and from there to screen (space)Canonical projection:

1000000000100001

z

y

x

In practice “ignore” z axis – use x and ycoordinates for screen coordinates

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Computer GraphicsComputer Graphics

Copyright 2004, A. Sheffer, UBC

Transformations

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Parallel Projection

Projectors are all parallel

Orthographic: Projectors perpendicular to projection plane

Oblique: Projectors not necessarily perpendicular to projection plane

ObliqueOrhtographic

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Perspective Projection

Viewing is from point at finite distanceWithout loss of generality:

Viewpoint at originViewing plane is z=d

Given P=(x,y,z) triangle similarity gives:

dzyy

dzxx

dy

zy

dx

zx

pppp

/ and

/ and ==⇒==

center of projection

3P

Z

projectors

projectionplane

),,( 1111 zyxP =

2P

31pp xx =

2px

dz=

X

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In matrix notation with homogeneous coordinates:

In Euclidean coordinates:

P singular: det(P)=0

Perspective Projection (cont’d)

P x y z x y zd

x y z zd

( , , , ) ( , , , )/

( , , , ).1 1

1 0 0 00 1 0 00 0 1 10 0 0 0

=

=

xz d

yz d

zz d

xz d

yz d

d x y dp p/,

/,

/ /,

/, ( , , ).

=

=

transformations

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Copyright 2004, A. Sheffer, UBC

Transformations

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Perspective Projection (cont’d)

What is (if any) is the difference between:Moving projection planeMoving viewpoint (center of projection)?

camtrans

Z

x

Z

x

Z

x

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Perspective Warp

P not invertible – can’t get depth (order) backIdea:

Warp viewing frustum (world)Then use parallel projection

Z

x

z=α z=d

Z

x

z=0 z=d

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Perspective Warp

Matrix formulation

Preserves relative depth (third coordinate)What does mean?

( , , , ) , ,( ) ,x y z dd d

dd

x y z dd

zd

1

1 0 0 00 1 0 00 0 1

0 0 0−−−

=−−

α

αα

αα

( , , )/

,/

,x y z xz d

yz d

dd zp p p =−

−

2

1α

α

0=α

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Copyright 2004, A. Sheffer, UBC

Transformations

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Perspective Warp

Warp modifies planes

warped to:

What does warp do to other objects? Spheres?

Aëdxp +Bëdyp +D(ëà d)zp +Cëd2 +Dd2 = 0

Ax+By+Cz+D= 0

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Another Transformations Quiz

What does each transformation preserve?

lines parallellines

distance angles normals convexity conics

scaling

rotation

translation

shear

perspective