1. Transformations

2. Last Time? Ray representation Generating rays from eye point / camera
orthographic camera
perspective camera
Find intersection point & surface normal
Primitives:
spheres, planes, polygons, triangles, boxes

3. Assignment 0 – main issues
Respect specifications!
Don’t put too much in the main function
Use object-oriented design
Especially since you will have to build on this code
Perform good memory management
Use new and delete
Avoid unnecessary temporary variables
Use enough precision for random numbers
Sample a distribution using cumulative probability

4. Outline Intro to Transformations Classes of Transformations
Representing Transformations
Combining Transformations
Transformations in Modeling
Adding Transformations to our Ray Tracer

5. 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

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

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

8. Outline Intro to Transformations Classes of Transformations
Representing Transformations
Combining Transformations
Transformations in Modeling
Adding Transformations to our Ray Tracer

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

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

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

12. 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

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

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

15. Perspective Projection

16. General (free-form) transformation
Does not preserve lines
Not as pervasive, computationally more involved
Won’t be treated in this course
From Sederberg and Parry, Siggraph 1986

17. Outline Intro to Transformations Classes of Transformations
Representing Transformations
Combining Transformations
Transformations in Modeling
Adding Transformations to our Ray Tracer

18. 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

19. 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

20. Translation in homogenous coordinates
x' = ax + by + c
y' = dx + ey + f
Affine formulation
Homogeneous formulation x' y‘ 1
a b
d e
0 0 c f 1 x y 1 x' y'
a b
d e x y c f = = +
p' = M p t
p' = M p

21. 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 =

22. 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

23. 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 =

24. 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 =

25. 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 =

26. 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

27. Storage Often, w is not stored (always 1)
Needs careful handling of direction vs. point
Mathematically, the simplest is to encode directions with w=0
In terms of storage, using a 3-component array for both direction and points is more efficient
Which requires to have special operation routines for points vs. directions

28. Outline Intro to Transformations Classes of Transformations
Representing Transformations
Combining Transformations
Transformations in Modeling
Adding Transformations to our Ray Tracer

29. 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!

30. 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)

31. 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 = =

32. Today Intro to Transformations Classes of Transformations
Representing Transformations
Combining Transformations
Transformations in Modeling
Adding Transformations to our Ray Tracer

33. Transformations in Modeling
Position objects in a scene
Change the shape of objects
Create multiple copies of objects
Projection for virtual cameras
Animations

34. Scene Description Scene Materials (much more next week) Camera Lights
Background
Objects

35. Simple Scene Description File
OrthographicCamera {
center
direction
up 0 1 0
size 5 }
Lights {
numLights 1
DirectionalLight {
direction
color } }
Background { color }
Materials {
numMaterials <n>
<MATERIALS> }
Group {
numObjects <n>
<OBJECTS> }

36. Class Hierarchy
Object3D
Cone
Sphere
Cylinder
Triangle
Plane
Group

37. Why is a Group an Object3D?
Logical organization of scene

38. Ray-group intersection
Recursive on all sub-objects

39. Simple Example with Groups
numObjects 3
Box { <BOX PARAMS> }
Box { <BOX PARAMS> } }
numObjects 2
Sphere { <SPHERE PARAMS> }
Sphere { <SPHERE PARAMS> } } }
Plane { <PLANE PARAMS> } }

40. Adding Materials Group { numObjects 3 Material { <BLUE> }
Box { <BOX PARAMS> }
Box { <BOX PARAMS> } }
numObjects 2
Material { <BROWN> }
Material { <GREEN> }
Material { <RED> }
Sphere { <SPHERE PARAMS> }
Material { <ORANGE> }
Sphere { <SPHERE PARAMS> } } }
Material { <BLACK> }
Plane { <PLANE PARAMS> } }

41. Adding Transformations

42. Class Hierarchy with Transformations
Object3D
Cone
Cylinder
Group
Sphere
Plane
Transform
Triangle

43. Why is a Transform an Object3D?
To position the logical groupings of objects within the scene

44. Simple Example with Transforms
Group {
numObjects 3
Transform {
ZRotate { 45 }
Box { <BOX PARAMS> }
Box { <BOX PARAMS> } } }
Translate { }
numObjects 2
Box { <BOX PARAMS> } }
Sphere { <SPHERE PARAMS> }
Sphere { <SPHERE PARAMS> } } } }
Plane { <PLANE PARAMS> } }

45. Nested Transforms p' = T ( S p ) = TS p same as Translate Translate
Scale
Scale
same as
Sphere
Sphere
Transform {
Translate { }
Scale { }
Sphere {
center 0 0 0
radius 1 } } }
Transform {
Translate { }
Scale { }
Sphere {
center 0 0 0
radius 1 } }