Transformations - Technion – Israel Institute of …webcourse.cs.technion.ac.il/.../ho/WCFiles/Transformations-bw.pdf · Geometric Transformations Transformation τis one-to-one

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,

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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?

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Applications

� Viewing

� Modeling

� Animation

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� Animation

� Image manipulation

Scaling

� V = (vx,vy) – vector in XY plane

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

( )( ) ( )yyxx

sssvsvVS yx ,

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( ) ( )yyxx svsvVS ,=

Scaling� Matrix form:

( )( ) ( ) ( )yyxx

sssvsv

svvVS yx ,

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� Independent in x and y

Non uniform scaling is allowed

� What is the meaning of scaling by zero?

Rotation

� Polar form:

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� Polar form:

V=(vx,vy)=(r cosα, r sinα)

� Rotating V anti-clockwise by θ to W:

)cossinsincos

,sinsincoscos(

))sin(),cos((

θαθαθαθα

θαθα

Rotation

� Matrix form:

θθθθ

cossin

sincos

cossin

sincos)sin,cos(

)cossinsincos

,sinsincoscos(

θαθαθαθα

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� Rotation operator R with parameter θ at the origin:

θθθθ

cossin

sincosR

Rotation Properties

� Rθ is orthogonal

� R-θ - rotation by -θ is

( ) ( )TRR θθ =−1

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� R - rotation by -θ is

cossin

sincos),()(

−− =

−= θθ

θθθθ

RvvVR yx

Translation

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� 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

h vvvvvV ==

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� Conversion (projection) from

homogeneous space to

Euclidean:

� In homogeneous

coordinates:

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

vvvV ,),(

Translation

� Using homogeneous coordinates, the translation

operator may be expressed as:

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)1,,()(),(

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

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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

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Rotate around origin by θ

Translate back

Transformation Composition

=−−

0010sincos001

),(),( ppppTRT yxyx

θθθθ

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−− 1

0cossin-

yxyx pppp

Transformations Quiz

� What do these

transformations do?

� And these homogeneous

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� 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?

Rotate by Shear

� Shear

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

1011sincos zxθθ

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cossin-

sincos

xxyzzxy

θθθθ

Rotate by Shear (cont.)

� Solve for x, y, z :

cos θ = 1+xy = yz+1

sin θ = −y = z+xyz+x

cossin-

sincos

xxyzzxy

θθθθ

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� 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

θθθθ sincos0secsin1

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� Why use it?

� Is it convenient?

� What happens when θ → π/2 ?

θθθθ

cossin-

sincos

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 ):

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� When is this rotation approximation useful?

� App. rotation matrix

� Matrix has to be orthogonal & normalized

Rotation Approximation

θθθAR

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� Matrix has to be orthogonal & normalized

� ARθ is orthogonal

� ARθ not normalized

�Normalize

1 θθ

θθ +=

−= cAR

3D Transformations

� All 2D transformations extend to 3D

� In homogeneous coordinates:

Scale Translate Rotate (around z axis)

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�� QuestionQuestion: What is ? ?

00cossin

00sincos

),,(),,( θθθθ

S zyxzyx

θyR transformations

3D Transformations

� Questions (commutativity):

Scaling: Is S1S2 = S2S1?

Translation: Is T1T2 = T2T1?

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Rotation: Is R1R2 = R2R1?

Example: Arbitrary Rotation

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� Problem:

Given two orthonormal coordinate systems XYZ and UVW ,

Find a transformation from one to the other.

Arbitrary Rotation

� Answer: Transformation matrix R

whose rows are U, V, W:

� Proof:

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� Similarly R(Y) = V & R(Z) = W

)0,0,1()(

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

mapping from UVW to XYZ

� E.g.

Arbitrary Rotation (cont.)

uuuUR yyy

=− ),,()(1

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� Comment: Used for mapping between XY plane and

arbitrary plane (why is this useful!?)

wvuuuuUR

yyyzyx

)0,0,1(

)0,0,(

),,()(

� 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 −+=

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

� Conclusion: polyhedron (polygon) is affinely transformed

by transforming its vertices

)()1()(

))1(()(

))1(())((

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

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Canonical projection:

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

coordinates for screen coordinates

ObjectObjectspacespace

ScreenScreenspacespace

Parallel Projection

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Parallel Projection

� Projectors are all parallel

Orthographic: Projectors perpendicular to projection plane

Axonometric: Rotation + translation + orthographic projection

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Parallel Projection� Projectors are all parallel

Oblique projection: Projectors not necessarily perpendicular to

projection plane

transformations

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� When can the oblique projection used?

What is Wrong in this Korean Picture?

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Perspective Projection

� Viewing is from point at a

finite distance

� Without loss of generality:

Viewpoint at origin

projection

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Viewing plane is z=d

� Given P=(x, y, z), triangle

similarity gives:

/ and ==⇒==

center of

projection

projectors

x1p= x3

p1= (x1, y1, z1)

� In matrix notation with homogeneous coordinates:

Perspective Projection (cont’d)

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

dzyxzyxP

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� In Euclidean coordinates:

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

).,,(,/

dyxddz

transformations

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

Moving the projection plane

camtrans

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Moving the viewpoint (center of projection)?

Examples of Perspective projection

Perspective could be defined with

one, two or three focal points:

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The same model in

Orthographic view Perspective view

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The same model with

Small perspective warp Large perspective warp

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More than Three Points’ Perspective!?

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

� Idea:

Warp the viewing frustum (and the entire world)

Then use parallel projection

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Question: Why

do we require

the depth

information?

Perspective Warp

� Matrix formulation

dzyx ,

)1,,,(α

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� This matrix maps the x and y coordinates much like

the perspective map in slide 34.

−−

− dd

( )( )

−−

xzyx ppp α

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!

( )( )

−−

xzyx ppp α

camtrans

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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

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� 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

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� A map that preserves angles is called conformal.

� A map that preserves distances is called an Isometry.

translation

perspective

Everything is a matter of Perspective

Sidewalk Chalk PaintingsSidewalk Chalk Paintings

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Julian Julian BeeverBeever

Transformations - Technion – Israel Institute of …webcourse.cs.technion.ac.il/.../ho/WCFiles/Transformations-bw.pdf · Geometric Transformations Transformation τis one-to-one

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