1. Modeling Transformation

2. Overview 2D Transformation 3D Transformation
Basic 2D transformations
Matrix representation
Matrix Composition
3D Transformation
Basic 3D transformations
Same as 2D
Transformation Hierarchies
Scene graphs
Ray casting

3. Modeling Transformation
Specify transformations for objects
Allow definitions of objects in own coordinate systems
Allow use of object definition multiple times in a scene
Ref] Hearn & Baker

4. 2D Modeling Transformations

5. 2D Modeling Transformations

6. Basic 2D Transformations
Translation
x’ = x + tx
y’ = y + ty
Scale
x’ = x * sx
y’ = y * sy
Shear
x’ = x + hx * y
y’ = y + hy * x
Rotation
x’ = x * cosq - y * sinq
y’ = x * sinq + y * cosq

7. Matrix Representation
Represent 2D transformation by a matrix
Multiply matrix by column vector
Apply transformation to point

8. Matrix Representation
Transformations combined by multiplication
Matrices are a convenient and efficient way to represent a sequence of transformations

9. 2 x 2 Matrices
What types of transformations can be represented with a 2 x 2 matrix?
2D Identity
2D Scale around (0, 0)

10. 2 x 2 Matrices
What types of transformations can be represented with a 2x 2 matrix?
2D Rotate around (0, 0)
2D Shear
shx = 2

11. 2 x 2 Matrices
What types of transformations can be represented with a 2x 2 matrix?
2D Mirror (=Reflection) over Y axis
2D Mirror over (0, 0)

12. Only linear 2D transformation can be presented with a 2 x 2 matrix
2 x 2 Matrices
What types of transformations can be represented with a 2x 2 matrix?
2D Translation ?
NO!
Only linear 2D transformation can be presented with a 2 x 2 matrix

13. Linear Transformations
Linear transformations are combinations of …
Scale
Rotation
Shear
Mirror (=Reflection)
Properties of linear transformations
Satisfies:
Origin maps to origin
Lines map to lines
Parallel line remain parallel
Ratios are preserved
Closed under composition

14. 2D Translation 2D Translation represented by a 3 x 3 matrix
Point represented with homogeneous coordinate

15. Homogeneous Coordinates
Add a 3rd coordinate to every 2D point
(x, y, w) represents a point at location (x/w, y/w)
(x, y, 0) represents a point at infinity
(0, 0, 0) is not allowed

16. Basic 2D Transformations
Basic 2D Transformations as 3 x 3 matrices

17. Affine Transformations
Affine Transformations are combinations of …
Linear transformation
Translation
Properties of affine transformations
Origin does not necessarily map to origin
Lines map to lines
Parallel line remain parallel
Ratios are preserved
Closed under composition

18. Transformation : Rigid-Body / Euclidean Transforms
Rigid Body Transformation (=rigid motion)
A transform made up of only translation and rotation
Preserve the shape of the objects that they act on
Preserves distances
Preserves angles
Rigid / Euclidean
Identity
Translation
Rotation

19. Transformation : Similitudes / Similarity Transforms
Similarity Transformations
Preserves angles
Translation, Rotation,
Isotropic scaling
Similitudes
Rigid / Euclidean
Identity
Translation
Isotropic Scaling
Rotation

20. Transformation : Linear Transformations
Similitudes
Linear
Rigid / Euclidean
Scaling
Identity
Translation
Isotropic Scaling
Reflection
Rotation
Shear
Translation is not linear

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

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

23. Projective Transformations
Affine transformation
Projective warps
Properties of projective transformations
Origin does not necessarily map to origin
Lines map to lines
Parallel line do not necessarily remain parallel
Ratios are not preserved (but “cross-ratios” are)
Closed under composition

24. Cross-ratio
Given any four points A, B, C, and D, taken in order, define their cross ratio as
Cross Ratio (ABCD) =
cross ratio (ABCD) = cross ratio (A'B'C'D')

25. Matrix Composition
Transformations can be combined by matrix multiplication

26. Matrix Composition
Matrices are a convenient and efficient way to represent a sequence of transformations
General purpose representations
Hardware matrix multiply
Efficiency with premultiplication

27. Matrix Composition Be aware: order of transformations matter
Matrix multiplication is not commutative
“Global”
“Local”

28. 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)

29. Matrix Composition Rotate by q around arbitrary point (a, b)
M = T(a, b)* R(q) * T(-a, -b)
Translate (a, b) to the origin
do the rotation about origin
translate back
Scale by sx, sy around arbitrary point (a, b)
M = T(a, b)* S(sx, sy) * T(-a, -b)
do the scale about origin

30. Transformation Game

31. Overview 2D Transformation 3D Transformation
Basic 2D transformations
Matrix representation
Matrix Composition
3D Transformation
Basic 3D transformations
Same as 2D
Transformation Hierarchies
Scene graphs
Ray casting

32. 3D Transformations Same idea as 2D transformations
Homogeneous coordinates: (x, y, z, w)
4 x 4 transformation matrices

33. Basic 3D transformations

34. Basic 3D transformations

35. Non-commutative Composition : Order of Rotations
Order of Rotation Affects Final Position
X-axis  Z-axis
Z-axis  X-axis

36. Matrix Composition : Rotations in space
Rotation about an Arbitrary Axis
Rotate about some arbitrary a through angle
Rotate about the z axis through an angle
The axis a lies in the yz plane
Rotate about the x axis through an angle
The axis a coincident with the z axis
Rotate about the a axis through
Same as a rotation about the now coincident z axis
Reverse the second rotation about the x axis
Reverse the first rotation about the z axis
Returning the axis a to its original position z a y x

37. Transformation Hierarchies
Scene may have hierarchy of coordinate systems
Each level stores matrix representing transformation from parent’s coordinate system

38. Transformation Example 1
Well-suited for humanoid characters

39. Transformation Example 1
Ref] Princeton University

40. Transformation Example 2
An object may appear in a scene multiple times

41. Transformation Example 2

42. Summary Coordinate systems
World coordinates
Modeling coordinates
Representations of 3D modeling transformations
4 x 4 Matrices
Scale, rotate, translate, shear, projections, etc.
Not arbitrary warps
Composition of 3D transformations
Matrix multiplication (order matters)
Transformation hierarchies