Coordinate Transformations & Homogeneous Coordinates

CS-C3100 Fall 2016 – Lehtinen

Coordinate

Transformations & Homogeneous

Coordinates

CS-C3100 (Introduction to) Computer Graphics Jaakko Lehtinen

Lots of slides from Frédo Durand

CS-C3100 Fall 2016 – Lehtinen 2

Outline• Intro to Transformations • Useful Classes of Transformations • Representing Transformations,

Homogeneous Coordinates • Combining Transformations • Hardcore: Transforming normals


Modeling/Viewing Pipeline Recap

3

1.Model the geometry • Here a triangle mesh. • Also, specify materials.



4

1.Model the geometry • Here a triangle mesh. • Also, specify materials.

Object coordinates



5

1.Model the geometry 2.Place the objects in

world space • Each object has its own

object space • Only one world space

Object coordinates World coordinates

Object Another Object

World Origin



6


world space 3.Pick viewing position

and direction

Object coordinates World coordinates View coordinates

Camera position and orientation



7


world space 3.Pick viewing

position and direction 4.Transform objects to

view space and project to image plane • Compute shading

Object coordinates World coordinates View coordinates Image coordinates


Modeling/Viewing Summary

Object coordinates

World coordinates

View coordinates

Image coordinates

✔

Today we’ll look at the transformations that take us between these coordinates.


Questions?

9


What is a Transformation?• A function that maps point x to point x':

Applications: animation, deformation, viewing, projection, real-time shadows, …

From Sederberg and Parry, Siggraph 1986


Simple Transformations

?


Simple Transformations

• Can be combined • Are these operations invertible?

Yes, except scale = 0


Questions?

13


Outline• Intro to Transformations • Useful Classes of Transformations

– This is the stuff you use for most modeling/viewing • Representing Transformations,

Homogeneous Coordinates • Combining Transformations • Transforming Normals


Rigid-Body / Euclidean Transforms

• What properties are preserved?

TranslationRotation

Rigid / Euclidean

Identity


Rigid-Body / Euclidean Transforms

• Preserves distances • Preserves angles

TranslationRotation

Rigid / Euclidean

Identity

• Preserves angles • “The shapes are the same”, just

at different scale, orientation and location


Similitudes / Similarity Transforms

TranslationRotation

Rigid / Euclidean

Similitudes

Isotropic ScalingIdentity


Linear Transformations

TranslationRotation

Rigid / EuclideanLinear

Similitudes

Isotropic ScalingIdentity

Scaling

Shear

Reflection



• L(p + q) = L(p) + L(q) • L(ap) = a L(p)

TranslationRotation


Similitudes

Isotropic Scaling

Scaling

Shear

ReflectionIdentity



• L(p + q) = L(p) + L(q) • L(ap) = a L(p)

TranslationRotation


Similitudes

Isotropic Scaling

Scaling

Shear

ReflectionIdentity

???



• L(p + q) = L(p) + L(q) • L(ap) = a L(p)

TranslationRotation


Similitudes

Isotropic Scaling

Scaling

Shear

ReflectionIdentity

???

Translation is not linear: f(p) = p+t

f(ap) = ap+t ≠ a(p+t) = a f(p) f(p+q) = p+q+t ≠ (p+t)+(q+t) = f(p) + f(q)


Affine Transformations

• What is preserved..?

TranslationRotation


Similitudes

Isotropic Scaling

Scaling

Shear

ReflectionIdentity

Affine


Affine Transformations

• Preserves parallel lines

TranslationRotation


Similitudes

Isotropic Scaling

Scaling

Shear

ReflectionIdentity

Affine


Projective Transformations• Preserves lines: lines remain lines

(planes remain planes in 3D)

TranslationRotation


Affine

(Planar) Projective

Similitudes

Isotropic Scaling

Scaling

Shear

Reflection

Perspective

Identity

• What’s with the hierarchy? Why have we grouped types of transformations with and within each other?


What’s so nice about these?

TranslationRotation

Rigid / Euclidean

LinearAffine(Planar) Projective

Similitudes

Isotropic Scaling

Scaling

ShearReflection

Perspective

Identity

• They are closed under concatenation – Means e.g. that an affine transformation followed by

another affine transformation is still an affine transformation

– Same for every subgroup, e.g. rotations, translations..

• Very convenient! • Projections are the

most general – Others are its

special casesCS-C3100 Fall 2016 – Lehtinen 26

What’s so nice about these?

TranslationRotation

Rigid / Euclidean

LinearAffine(Planar) Projective

Similitudes

Isotropic Scaling

Scaling

ShearReflection

Perspective

Identity


Name-dropping

27

• Fancy name: Group Theory • Remember algebra?

– A group is a set S with an operation f that takes two elements of S and produces a third: (and some other axioms)

• These transformations are group(s) and subgroups – The transformations are the set S,

concatenation of transformations is f

s, t ⇥ S, f(s, t) = u � u ⇥ S


Transforms are Groups

28

• Why is this useful? – You can represent any number of

successive transformations by a single compound transformation

• Example – The object-to-world transformation,

the world-to-view transformation, and the perspective projection (view-to-image) can all be folded into a single projective object-to-image transformation

– (OpenGL: Modelview, projection)

Disclaimer:Not ANY

transformation, but the types

just introduced

Object coordinates World coordinates View coordinates Image coordinates


Questions?

29


More Complex Transformations..

30

• ...can be built out of these, e.g.

• Skinning – Blending of affine

transformations – We’ll do this later! – And you will code

it up! :) Ilya Baran


More Complex Transformations

31

• Harmonic coordinates (link to paper) – Object enclosed in simple “cage”, each object point

knows the influence each cage vertex has on it – Deform the cage, and the object moves!

Pix

ar

http://dl.acm.org/citation.cfm?id=1276466


Questions?





Interlude: ImmersiveMath.com

34

• If your matrices and vectors are a little rusty, and even if they aren’t, read the first chapters from this really neat interactive linear algebra “book”!

http://immersivemath.com


How are Linear Transforms Represented?

Linear transformations include rotation, scaling,

reflection, and shear



x' = ax + by y' = dx + ey





x' y'

a b d e=

x y

p' = M p






x' y'

a b d e=

x y

p' = M p




Note that the origin always stays fixed.


How are Affine Transforms Represented?

x' = ax + by + c y' = dx + ey + f

x' y'

a b d e

c f=

x y

+

p' = M p + t

Affine transformations include all linear

transformations, plus translation


Can we use matrices for affine transforms?

x' = ax + by + c y' = dx + ey + f

p' = M p ?

x' y'

a b d e=

x y

p' = M p


The Homogeneous Coordinate Trickx' = ax + by + c y' = dx + ey + f

x y 1

x' y'

a b d e

c f=

x y

+

p' = M p + t

Affine formulation Homogeneous formulation

This is whatwe want:


x' = ax + by + c y' = dx + ey + f

x' y‘ 1

a b d e 0 0

c f 1

=x y 1

p' = M p

x' y'

a b d e

c f=

x y

+

p' = M p + t

Affine formulation Homogeneous formulation

This is whatwe want:

The Homogeneous Coordinate Trick


Homogeneous Coordinates• Add an extra dimension

– in 2D, we use 3-vectors and 3 x 3 matrices – In 3D, we use 4-vectors and 4 x 4 matrices – Makes affine transformations linear in one higher

dimension (can be represented by a matrix) • Each point has an extra value, w

– You can think of it as “scale” – For all transformations except

perspective, you can just set w=1 and not worry about it

x' y‘ 1

a b d e 0 0

c f 1

=x y 1


Homogeneous Coordinates

• If we multiply a homogeneous point by an affine matrix, w is unchanged – “Affine matrix” means the last row is (0 0 0 1) – This form encodes all possible affine transformations! – What happens when you multiply an affine matrix with

another affine matrix?

x' y' z' 1

=

x y z 1

a e i 0

b f j 0

c g k 0

d h l 1


More on w

45

• As mentioned, you can think of it as “scale”. • Let’s see:

x' y' 1

a b d e 0 0

c f 1

=x y 1

? a b d e 0 0

c f 1

=2x 2y 2


More on w

46


x' y' 1

a b d e 0 0

c f 1

=x y 1

(DUH!)2x' 2y' 2

a b d e 0 0

c f 1

=2x 2y 2


More on w

47


x' y' 1

a b d e 0 0

c f 1

=x y 1

2x' 2y' 2

a b d e 0 0

c f 1

=2x 2y 2

(DUH!)It’s tempting to

divide by w, because this will get us back to

the result above.


“Projective Equivalence” in 1D

48

w=0

w=1

w=2

w=3

w

x (0,0)

(x,w)


Projective Equivalence in 1D

49

• We can get into a “canonical form” by dividing by w. This projects (x, w) onto the line w=1 yielding (x/w, 1).

w=0

w=1

w=2

w=3

w

x (0,0)

(x,w)

(x/w,1)


Projective Equivalence in 1D

50

• We can get into a “canonical form” by dividing by w. This projects (x, w) onto the line w=1 yielding (x/w, 1).

• We say that all points on dashed line are identical becausethey project to the same point on the line w=1.

w=0

w=1

w=2

w=3

w

x (0,0)

(x,w)

(x/w,1)

• More mathematically, all non-zero scalar multiples of a point are considered identical • So, in fact, we are saying that points in ND are

identified with lines through the origin in (N+1)D


Projective Equivalence

51

x y z w

ax ay az aw

a != 0

=x/w y/w z/w 1

=w !=0


Homogeneous Visualization in 2D• Divide by w to normalize (project)

w = 1

w = 2

(0, 0, 1) = (0, 0, 2) = …(7, 1, 1) = (14, 2, 2) = …(4, 5, 1) = (8, 10, 2) = …

(0,0,0)


Homogeneous Visualization in 2D• Divide by w to normalize (project) • w = 0?

w = 1

w = 2

(0, 0, 1) = (0, 0, 2) = …(7, 1, 1) = (14, 2, 2) = …(4, 5, 1) = (8, 10, 2) = …

Points at infinity (directions)

(0,0,0)


Projective Equivalence – Why?

54

• For affine transformations, adding w=1 in the end proved to be convenient.

• The real showpiece is perspective.


Projective Equivalence – Why?

55

• For affine transformations, adding w=1 in the end proved to be convenient.

• The real showpiece is perspective. – From the camera’s point of view, all 3D points on the

line fall on the same 2D coordinate in the image. – In that sense, all those 3D points are

“identical”.


Questions?

• Why bother with the extra dimension? – Because now translations

can be encoded in the matrix!


x' y' z' 1

Translate (tx, ty, tz)

x' y' z'

=

x y z 1

1 0 0 0

0 1 0 0

0 0 1 0

tx

ty

tz

1

Translate(c,0,0)

x

y

p p'

c


What’s Actually Happening?

58

x' = x + c

x x’

c



59

x' = x + c

c

(x,1) (x’,1)

Let’s first add the 2nd dimension w and set coordinates to 1. What does this mean..?


w=1

w=2

w=0(0,0)

x' = x + c

x

w


(x,1) (x’,1)

xx’


w=1

w=2

w=0(0,0)

x' = x + c

x

w

What’s Actually Happening?xx’


w=1

w=2

w=0(0,0)x

w

x' 1

1 c 0 1=

x 1

What’s Actually Happening?xx’



63

w=1

w=2

w=0(0,0)

x' 1

1 c 0 1=

x 1

It’s a 2D shear!

This line stays put, and origin stays still

x

w

x

w


Questions?


Scale (sx, sy, sz)• Isotropic (uniform)

scaling: sx = sy = sz

x' y' z' 1

=

x y z 1

sx

0 0 0

0 sy

0 0

0 0 sz

0

0 0 0 1

Scale(s,s,s)

x

p

p'

qq'

y


Rotation• About z axis

x' y' z' 1

=

x y z 1

cos θ sin θ

0 0

-sin θ cos θ

0 0

0 0 1 0

0 0 0 1

ZRotate(θ)

x

y

z

p

p'

θ


Rotation• About (kx, ky, kz), a unit

vector on an arbitrary axis (Rodrigues Formula)

x' y' z' 1

=

x y z 1

kxkx(1-c)+c kykx(1-c)+kzs kzkx(1-c)-kys

0

0 0 0 1

kxky(1-c)-kzs kyky(1-c)+c

kzky(1-c)+kxs 0

kxkz(1-c)+kys kykz(1-c)-kxs kzkz(1-c)+c

0

where c = cos θ & s = sin θ

Rotate(k, θ)

x

y

z

θ

k


Rotations and Matrices

68

• Rotations are represented by orthonormal matrices, i.e., matrices such that MTM = I – This implies det(M) = ±1 (why?) – Furthermore, to rule out reflections, require det(M)=1

• Rotations are a group of their own – Can you prove this based on the above properties?


Questions?


A Word of Warning

70

• In “regular” 3D, adding a displacement d to a point p is simple

• What about homogeneous coordinates?

p� = p + d =

�

⇤px + dx

py + dy

pz + dz

⇥

⌅

p� =

�

⇧⇧⇤

px

py

pz

1

⇥

⌃⌃⌅ +

�

⇧⇧⇤

dx

dy

dz

⇥

⌃⌃⌅?


A Word of Warning

71

• In “regular” 3D, adding a displacement d to a point p is simple


p� = p + d =

�

⇤px + dx

py + dy

pz + dz

⇥

⌅

0p� =

�

⇧⇧⇤

px

py

pz

1

⇥

⌃⌃⌅ +

�

⇧⇧⇤

dx

dy

dz

⇥

⌃⌃⌅ =

�

⇧⇧⇤

px + dx

py + dy

pz + dz

1

⇥

⌃⌃⌅

You can’t add homogeneous points to each

other like in 3D.


A Word of Warning

72

• In 3D, adding a displacement d to a point p is simple


p� = p + d =

�

⇤px + dx

py + dy

pz + dz

⇥

⌅

0p� =

�

⇧⇧⇤

px

py

pz

1

⇥

⌃⌃⌅ +

�

⇧⇧⇤

dx

dy

dz

⇥

⌃⌃⌅ =

�

⇧⇧⇤

px + dx

py + dy

pz + dz

1

⇥

⌃⌃⌅

w stays the same!

px' py' pz' 1

=

px py pz 1

1 0 0 0

0 1 0 0

0 0 1 0

dx

dy

dz

1

You can’t add homogeneous points to each

other like in 3D.


The Same for Scaling

73

a p =

�

⇧⇧⇤

a px

a py

a pz

a

⇥

⌃⌃⌅Remember projective

equivalence


The Same for Scaling

74

• You also leave w fixed.

a p =

�

⇧⇧⇤

a px

a py

a pz

1

⇥

⌃⌃⌅


Summary

75

• Affine n-D transformations are encoded by linear transformations in (n+1)D that move corresponding homogeneous points in the w=1 (hyper)plane – Technically, w=any constant works too, just have to

scale translation distances as well.

Important!


Questions?


Why Bother II: Perspective

w = 1

w = 2

77

• This picture gives away almost the whole story.


Perspective in 2D

78

• Camera at origin, looking along z, 90 degree f.o.v., “image plane” at z=1

z=1

z=0xz

p=(x,z)

p’=(x/z,1)

(0,0)

x=zx=-z


Perspective in 2D

79

z=1

z=0xz

p=(x,z)

p’=(x/z,1)

(0,0)

x=zx=-z

p� =

�

⇤x/z11

⇥

⌅The projected point in

homogeneous coordinates

(we just added w=1):


Perspective in 2D

80

z=1

z=0xz

p=(x,z)

p’=(x/z,1)

(0,0)

x=zx=-z

p� =

�

⇤x/z11

⇥

⌅ �

�

⇤xzz

⇥

⌅

Projectively equivalent


How do you do this with a matrix?

81

0

@x

z

z

1

A =

0

@ ?

1

A

0

@x

z

1

1

A


Perspective in 2D

82

z=1

z=0xz

p=(x,z)

p’=(x/z,1)

(0,0)

x=zx=-z

p� �

�

⇤xzz

⇥

⌅ =

�

⇤1 0 00 1 00 1 0

⇥

⌅

�

⇤xz1

⇥

⌅ ,

We’ll just copy z to w, and get the

projected point after homogenization!

�

⇧⇧⇤

x�

y�

z�

w�

⇥

⌃⌃⌅ =

�

⇧⇧⇤

1 0 0 00 1 0 00 0 1 00 0 1 0

⇥

⌃⌃⌅

�

⇧⇧⇤

xyz1

⇥

⌃⌃⌅


Extension to 3D

83

• Trivial:Just add another dimension y and treat it like x – z is the special one, it turns into w’

• Different fields of view and non-square image aspect ratios can be accomplished by simple scaling of the x and y axes.


Caveat

84

• These projections matrices work perfectly in the sense that you get the proper 2D projections of 3D points.

• However, since we are flattening the scene onto the z=1 plane, we’ve lost all information about the distance to camera. – Not a big deal for ray tracers, but GPUs need distances

for “Z buffering”, i.e., figuring out what is in front of what.


The “View Frustum” in 2D

85

• (In 3D this would be a truncated pyramid.)

z=1

z=far

z=0xz

p

(0,0)

z=near

z’=1

z’=0x’z’

p’q

q’

x’=-1 x’=1


The View Frustum in 2D

86

• We can transform the frustum by a modified projection in a way that makes it a square after projection and homogenization (division by w).

�

⇤x⇤

z⇤

w⇤

⇥

⌅ =

�

⇤1 0 00 f+n

f�n � 2⇥f⇥nf�n

0 1 0

⇥

⌅

�

⇤xz1

⇥

⌅ ,

xz

x’/w’

z’/w’

These z’/w’ values

go into the Z buffer.


Details in Handout in MyCourses

87

• “Understanding Projections and Homogenous Coordinates” – How you get the matrix from previous slide

• Fun demonstration: Print the following page out. Holding the paper as flat as possible, try to see if you can hold it at such an angle in front of you so that the grid looks like a square with a regular grid in it.



Cool Application Video• “Structure from Motion” -algorithms

– SfM if a branch of computer vision that tries to understand the 3D structure of the scene from pictures taken from different viewpoints.

– It’s all based on projective geometry and homogeneous coordinates.

– Examples: • http://phototour.cs.washington.edu/ • http://phototour.cs.washington.edu/findingpaths/

http://phototour.cs.washington.edu

http://phototour.cs.washington.edu/findingpaths/


Want to Know More?

90

• Take Ville Kyrki’s Machine Perception (AS-84.3126) to know more!



Phase 3: Profit• All these are linear in

homogeneous coordinates!

TranslationRotation


Affine

Projective

Similitudes

Isotropic Scaling

Scaling

Shear

Reflection

Perspective

Identity





How are transforms combined?

(0,0)(1,1)

(2,2)

(0,0)

(5,3)

(3,1)Scale(2,2) Translate(3,1)

TS =2 0

0 2

0 0

1 0

0 1

3 1

2 0

0 2

3 1=

Scale then Translate

Use matrix multiplication: p' = T ( S p ) = TS p

Caution: matrix multiplication is NOT commutative!

0 0 1 0 0 1 0 0 1


Non-commutative CompositionScale then Translate: p' = T ( S p ) = TS p

Translate then Scale: p' = S ( T p ) = ST p

(0,0)

(1,1)(4,2)

(3,1)

(8,4)

(6,2)

(0,0)(1,1)

(2,2)

(0,0)

(5,3)

(3,1)Scale(2,2) Translate(3,1)

Translate(3,1) Scale(2,2)


TS =2 0 0

0 2 0

0 0 1

1 0 0

0 1 0

3 1 1

ST =2 0

0 2

0 0

1 0

0 1

3 1

Non-commutative CompositionScale then Translate: p' = T ( S p ) = TS p

2 0 0

0 2 0

3 1 1

2 0

0 2

6 2

=

=

Translate then Scale: p' = S ( T p ) = ST p

0 0 1 0 0 1 0 0 1


Questions?





Normal• Surface Normal: unit vector that is locally

perpendicular to the surface


Why is the Normal important?• It's used for shading — makes things look 3D!

object color only Diffuse Shading


Visualization of Surface Normal

± x = Red ± y = Green ± z = Blue


How do we transform normals?

Object Space World Space

nOS

nWS


Transform Normal like Object?• translation? • rotation? • isotropic scale? • scale? • reflection? • shear? • perspective?


Transform Normal like Object?• translation? • rotation? • isotropic scale? • scale? • reflection? • shear? • perspective?


Similitudes

What class of transforms?

TranslationRotation


Affine

Projective

Similitudes

Isotropic Scaling

Scaling

Shear

Reflection

Perspective

IdentityTranslation

RotationIsotropic Scaling

IdentityReflection

a.k.a. Orthogonal Transforms


Transformation for shear and scale

Incorrect Normal

Transformation

Correct Normal

Transformation


More Normal Visualizations

Incorrect Normal Transformation Correct Normal Transformation


• Think about transforming the tangent plane to the normal, not the normal vector

So how do we do it right?

Original Incorrect Correct

nOS

Pick any vector vOS in the tangent plane, how is it transformed by matrix M?

vOSvWS

nWS

vWS = M vOS


Transform tangent vector vv is perpendicular to normal n:

nOST vOS = 0

nOST (M-1 M) vOS = 0

nWST = nOS

T (M-1)

(nOST M-1) (M vOS) = 0 (nOS

T M-1) vWS = 0

nWST vWS = 0

vWS is perpendicular to normal nWS:

nWS = (M-1)T nOS

nOS

vWS

nWS

vOS

Dot product


Digression

110

• The previous proof is not quite rigorous; first you’d need to prove that tangents indeed transform with M. – Turns out they do, but we’ll take it on faith here. – If you believe that, then the above formula follows.

nWS = (M-1)T nOS


Comment• So the correct way to transform normals is:

• But why did nWS = M nOS work for similitudes? • Because for similitude / similarity transforms,

(M-1)T =λ M • e.g. for orthonormal basis: M-1 = M T i.e. (M-1)T = M

nWS = (M-1)T nOS Sometimes denoted M-T


Connections• Not part of class, but cool

– “Covariant”: transformed by the matrix • e.g., tangent

– “Contravariant”: transformed by the inverse transpose • e.g., the normal • a normal is a “co-vector”

• Google “differential geometry” to find out more


Questions?

113


That’s All for Today

114

• Further Reading – Projection handout (MyCourses)

• Other Cool Stuff – Algebraic Groups – http://phototour.cs.washington.edu/ – http://phototour.cs.washington.edu/findingpaths/ – Free-form deformation of solid objects – Harmonic coordinates for character articulation

http://en.wikipedia.org/wiki/Group_%2528mathematics%2529

http://doi.acm.org/10.1145/15886.15903

http://phototour.cs.washington.edu

http://phototour.cs.washington.edu/findingpaths/

http://doi.acm.org/10.1145/15886.15903

http://doi.acm.org/10.1145/1276377.1276466

Documents

Coordinate Transformations & Homogeneous Coordinates