1. Transformations

2. Outline Assignment 0 Recap Intro to Transformations
Classes of Transformations
Representing Transformations
Combining Transformations
Change of Orthonormal Basis

3. Cool Results from Assignment 0
emmav
alexg
psoto
koi
scyudits

4. Notes on Assignments
Make sure you turn in a linux or windows executable (so we can test your program)
Don't use athena dialup
Collaboration Policy
Share ideas, not code
Ask questions during office hours, or
Tell us how much time you spent on each assignment

5. Quick Review of Last Week
Ray representation
Generating rays from eyepoint / camera
orthographic camera
perspective camera
Find intersection point & surface normal
Primitives:
spheres, planes, polygons, triangles, boxes

6. Outline Assignment 0 Recap Intro to Transformations
Classes of Transformations
Representing Transformations
Combining Transformations
Change of Orthonormal Basis

7. What is a Transformation?
Maps points (x, y) in one coordinate system to points (x', y') in another coordinate system
For example, IFS:
x' = ax + by + c
y' = dx + ey + f

8. Common Coordinate Systems
Object space
local to each object
World space
common to all objects
Eye space / Camera space
derived from view frustum
Screen space
indexed according to hardware attributes

9. Simple Transformations
Can be combined
Are these operations invertible?
Yes, except scale = 0

10. Transformations are used:
Position objects in a scene (modeling)
Change the shape of objects
Create multiple copies of objects
Projection for virtual cameras
Animations

11. Outline Assignment 0 Recap Intro to Transformations
Classes of Transformations
Rigid Body / Euclidean Transforms
Similitudes / Similarity Transforms
Linear
Affine
Projective
Representing Transformations
Combining Transformations
Change of Orthonormal Basis

12. Rigid-Body / Euclidean Transforms
Preserves distances
Preserves angles
Rigid / Euclidean
Identity
Translation
Rotation

13. Similitudes / Similarity Transforms
Preserves angles
Similitudes
Rigid / Euclidean
Identity
Translation
Isotropic Scaling
Rotation

14. Linear Transformations
Similitudes
Linear
Rigid / Euclidean
Scaling
Identity
Translation
Isotropic Scaling
Reflection
Rotation
Shear

15. Linear Transformations
L(p + q) = L(p) + L(q)
L(ap) = a L(p)
Similitudes
Linear
Rigid / Euclidean
Scaling
Identity
Translation
Isotropic Scaling
Reflection
Rotation
Shear

16. Affine Transformations
preserves parallel lines
Affine
Similitudes
Linear
Rigid / Euclidean
Scaling
Identity
Translation
Isotropic Scaling
Reflection
Rotation
Shear

17. Projective Transformations
preserves lines
Projective
Affine
Similitudes
Linear
Rigid / Euclidean
Scaling
Identity
Translation
Isotropic Scaling
Reflection
Rotation
Shear
Perspective

18. Perspective Projection

19. Outline Assignment 0 Recap Intro to Transformations
Classes of Transformations
Representing Transformations
Combining Transformations
Change of Orthonormal Basis

20. How are Transforms Represented?
x' = ax + by + c
y' = dx + ey + f x' y'
a b
d e x y c f = +
p' = M p t

21. Homogeneous Coordinates
Add an extra dimension
in 2D, we use 3 x 3 matrices
In 3D, we use 4 x 4 matrices
Each point has an extra value, w x' y' z' w' a e i m b f j n c g k o d h l p x y z w =
p' = M p

22. Homogeneous Coordinates
Most of the time w = 1, and we can ignore it
If we multiply a homogeneous coordinate by an affine matrix, w is unchanged x' y' z' 1 a e i b f j c g k d h l 1 x y z 1 =

23. Homogeneous Visualization
Divide by w to normalize (homogenize)
W = 0?
Point at infinity (direction)
(0, 0, 1) = (0, 0, 2) = …
w = 1
(7, 1, 1) = (14, 2, 2) = …
(4, 5, 1) = (8, 10, 2) = …
w = 2

24. Translate (tx, ty, tz)
Translate(c,0,0) y
Why bother with the extra dimension? Because now translations can be encoded in the matrix! p p' x c x' y' z' 1 x' y' z' 1 1 1 tx ty tz 1 x y z 1 =

25. Scale (sx, sy, sz) Isotropic (uniform) scaling: sx = sy = sz x' y' z'
Scale(s,s,s) y p'
Isotropic (uniform) scaling: sx = sy = sz p q' q x x' y' z' 1 sx sy sz 1 x y z 1 =

26. Rotation About z axis x' y' z' 1 cos θ sin θ -sin θ cos θ 1 1 x y z 1
ZRotate(θ) y p'
About z axis θ p x z x' y' z' 1
cos θ
sin θ
-sin θ
cos θ 1 1 x y z 1 =

27. Rotation About x axis: About y axis: x' y' z' 1 1 cos θ sin θ -sin θ
cos θ
sin θ
-sin θ
cos θ 1 x y z 1 = x' y' z' 1
cos θ
-sin θ 1
sin θ
cos θ 1 x y z 1 =

28. Rotation
Rotate(k, θ) y
About (kx, ky, kz), a unit vector on an arbitrary axis (Rodrigues Formula) θ k x z x' y' z' 1
kxkx(1-c)+c
kykx(1-c)+kzs
kzkx(1-c)-kys
kzkx(1-c)-kzs
kzkx(1-c)+c
kzkx(1-c)-kxs
kxkz(1-c)+kys
kykz(1-c)-kxs
kzkz(1-c)+c 1 x y z 1 =
where c = cos θ & s = sin θ

29. Outline Assignment 0 Recap Intro to Transformations
Classes of Transformations
Representing Transformations
Combining Transformations
Change of Orthonormal Basis

30. How are transforms combined?
Scale then Translate
(5,3)
Scale(2,2)
(2,2)
Translate(3,1)
(1,1)
(3,1)
(0,0)
(0,0)
Use matrix multiplication: p' = T ( S p ) = TS p 1 1 1 3 1 1 2 2 1 2 2 3 1
TS = =
Caution: matrix multiplication is NOT commutative!

31. Non-commutative Composition
Scale then Translate: p' = T ( S p ) = TS p
(5,3)
Scale(2,2)
(2,2)
Translate(3,1)
(1,1)
(3,1)
(0,0)
(0,0)
Translate then Scale: p' = S ( T p ) = ST p
(8,4)
Translate(3,1)
(4,2)
Scale(2,2)
(1,1)
(6,2)
(3,1)
(0,0)

32. Non-commutative Composition
Scale then Translate: p' = T ( S p ) = TS p 1 1 3 1 2 2 1 2 2 3 1
TS = =
Translate then Scale: p' = S ( T p ) = ST p 2 1 2 1 1 1 3 1 1 2 2 6 2
ST = =

33. Outline Assignment 0 Recap Intro to Transformations
Classes of Transformations
Representing Transformations
Combining Transformations
Change of Orthonormal Basis

34. Review of Dot Product b a

35. Change of Orthonormal Basis
Given:
coordinate frames xyz and uvn
point p = (x,y,z)
Find:
p = (u,v,n) y p x v v u u y x y v p u n x z

36. Change of Orthonormal Basisy y
y . u v v
y . v u u
x . v x x n
x . u z x y z =
(x . u) u
(y . u) u
(z . u) u +
(x . v) v
(y . v) v
(z . v) v +
(x . n) n
(y . n) n
(z . n) n

37. Change of Orthonormal Basisx y z =
(x . u) u
(y . u) u
(z . u) u +
(x . v) v
(y . v) v
(z . v) v +
(x . n) n
(y . n) n
(z . n) n
Substitute into equation for p:
p = (x,y,z) = x x + y y + z z
p =
x [
y [
z [
(x . u) u
(y . u) u
(z . u) u +
(x . v) v
(y . v) v
(z . v) v +
(x . n) n
(y . n) n
(z . n) n
] + ]

38. Change of Orthonormal Basis
p =
x [
y [
z [
(x . u) u
(y . u) u
(z . u) u +
(x . v) v
(y . v) v
(z . v) v +
(x . n) n
(y . n) n
(z . n) n
] + ]
Rewrite:
p = [
x (x . u)
x (x . v)
x (x . n) +
y (y . u)
y (y . v)
y (y . n) +
z (z . u)
z (z . v)
z (z . n)
] u +
] v +
] n

39. Change of Orthonormal Basis
p = [
x (x . u)
x (x . v)
x (x . n) +
y (y . u)
y (y . v)
y (y . n) +
z (z . u)
z (z . v)
z (z . n)
] u +
] v +
] n
p = (u,v,n) = u u + v v + n n
Expressed in uvn basis: u v n =
x (x . u)
x (x . v)
x (x . n) +
y (y . u)
y (y . v)
y (y . n) +
z (z . u)
z (z . v)
z (z . n)

40. Change of Orthonormal Basisu v n =
x (x . u)
x (x . v)
x (x . n) +
y (y . u)
y (y . v)
y (y . n) +
z (z . u)
z (z . v)
z (z . n)
In matrix form:
where: u v n ux vx nx uy vy ny uz vz nz x y z
ux = x . u =
uy = y . u
etc.

41. Change of Orthonormal Basisu v n ux vx nx uy vy ny uz vz nz x y z x y z =
= M
What's M-1, the inverse? x y z xu yu zu xv yv zv xn yn zn u v n
ux = x . u = u . x = xu =
M-1 = MT

42. Adding Transformations to the Ray Caster (Assignment 2)
Next Time:
Adding Transformations to the Ray Caster (Assignment 2)