1. Affine Geometry

2. Geometric Programming
A way of handling geometric entities such as vectors, points, and transforms.
Traditionally, computer graphics packages are implemented using homogeneous coordinates.
We will review affine geometry and coordinate-invariant geometric programming.
Because of historical reasons, it has been called “coordinate-free” geometric programming

3. Example of coordinate-dependence
Point p
Point q
What is the “sum” of these two positions ?

4. If you assume coordinates, …
p = (x1, y1)
q = (x2, y2)
The sum is (x1+x2, y1+y2)
Is it correct ?
Is it geometrically meaningful ?

5. If you assume coordinates, …
p = (x1, y1)
(x1+x2, y1+y2)
q = (x2, y2)
Origin
Vector sum
(x1, y1) and (x2, y2) are considered as vectors from the origin to p and q, respectively.

6. If you select a different origin, …
p = (x1, y1)
(x1+x2, y1+y2)
q = (x2, y2)
Origin
If you choose a different coordinate frame, you will get a different result

7. Vector and Affine Spaces
Vector space
Includes vectors and related operations
No points
Affine space
Superset of vector space
Includes vectors, points, and related operations

8. Points and Vectors Point q vector (p-q) Point p
A point is a position specified with coordinate values.
A vector is specified as the difference between two points.
If an origin is specified, then a point can be represented by a vector from the origin.
But, a point is still not a vector in coordinate-free concepts.

9. Vector spaces A vector space consists of
Set of vectors, together with
Two operations: addition of vectors and multiplication of vectors by scalar numbers
A linear combination of vectors is also a vector

10. Affine Spaces An affine space consists of
Set of points, an associated vector space, and
Two operations: the difference between two points and the addition of a vector to a point

11. Coordinate-Free Geometric Operations
Addition
Subtraction
Scalar multiplication
Linear combination
Affine combination

12. Addition p + w u + v v w u p u + v is a vector p + w is a point
u, v, w : vectors
p, q : points

13. Subtraction p p - w p - q -w u u - v v q p u - v is a vector
p - q is a vector
p - w is a point
u, v, w : vectors
p, q : points

14. Scalar Multiplication
scalar • vector = vector
1 • point = point
0 • point = vector
c • point = (undefined) if (c≠0,1)

15. Linear Combination A linear space is spanned by a set of bases
Any point in the space can be represented as a linear combination of bases

16. Affine Combination

17. Example (p + q) / 2 : midpoint of line pq
(p + q) / 3 : Can you find a geometric meaning ?
(p + q + r) / 3 : center of gravity of ∆pqr
(p/2 + q/2 – r) : a vector from r to the midpoint of q and p

18. Affine Frame
A frame is defined as a set of vectors {vi | i=1, …, N} and a point o
Set of vectors {vi} are bases of the associate vector space
o is an origin of the frame
N is the dimension of the affine space
Any point p can be written as
Any vector v can be written as

19. Summary

20. Geometric Transformations

21. Transformations Linear transformations Rigid transformations
Local moving frame
Linear transformations
Rigid transformations
Affine transformations
Projective transformations T
Global reference frame

22. Linear Transformations
A linear transformation T is a mapping between vector spaces
T maps vectors to vectors
linear combination is invariant under T
In 3D-spaces, T can be represented by a 3x3 matrix

23. Examples of Linear Transformations
2D rotation
2D scaling

24. Examples of Linear Transformations
2D shear
Along X-axis
Along Y-axis

25. Examples of Linear Transformations
2D reflection
Along X-axis
Along Y-axis

26. Properties of Linear Transformations
Any linear transformation between 3D spaces can be represented by a 3x3 matrix
Any linear transformation between 3D spaces can be represented as a combination of rotation, shear, and scaling

27. Composing Transformations
Suppose we want,
We have to compose two transformations

28. Composing Transformations
Matrix multiplication is not commutative
Translation
followed by
rotation
Translation
followed by
rotation

29. Composing Transformations
R-to-L : interpret operations w.r.t. fixed coordinates
L-to-R : interpret operations w.r.t local coordinates
(Column major convention)

30. Changing Bases
Linear transformations as a change of bases

31. Changing Bases
Linear transformations as a change of bases

32. Affine Transformations
An affine transformation T is an mapping between affine spaces
T maps vectors to vectors, and points to points
T is a linear transformation on vectors
affine combination is invariant under T
In 3D-spaces, T can be represented by a 3x3 matrix together with a 3x1 translation vector

33. Homogeneous Coordinates
Any affine transformation between 3D spaces can be represented by a 4x4 matrix
Affine transformation is linear in homogeneous coordinates

34. Projective Spaces Homogeneous coordinates
(x, y, z, w) = (x/w, y/w, z/w, 1)
Useful for handling perspective projection
But, it is algebraically inconsistent !!

35. Examples of Affine Transformations
2D rotation
2D scaling

36. Examples of Affine Transformations
2D shear
2D reflection

37. Examples of Affine Transformations
2D translation

38. Examples of Affine Transformations
2D transformation for vectors
Translation is simply ignored

39. Examples of Affine Transformations
3D rotation

40. Pivot-Point Rotation
Rotation with respect to a pivot point (x,y)

41. Fixed-Point Scaling
Scaling with respect to a fixed point (x,y)

42. Scaling Direction
Scaling along an arbitrary axis

43. Properties of Affine Transformations
Any affine transformation between 3D spaces can be represented as a combination of a linear transformation followed by translation
An affine transf. maps lines to lines
An affine transf. maps parallel lines to parallel lines
An affine transf. preserves ratios of distance along a line
An affine transf. does not preserve absolute distances and angles

44. Review of Affine Frames
A frame is defined as a set of vectors {vi | i=1, …, N} and a point o
Set of vectors {vi} are bases of the associate vector space
o is an origin of the frame
N is the dimension of the affine space
Any point p can be written as
Any vector v can be written as

45. Changing Frames
Affine transformations as a change of frame

46. Changing Frames
Affine transformations as a change of frame

47. Changing Frames
In case the xyz system has standard bases

48. Rigid Transformations
A rigid transformation T is a mapping between affine spaces
T maps vectors to vectors, and points to points
T preserves distances between all points
T preserves cross product for all vectors (to avoid reflection)
In 3-spaces, T can be represented as

49. Rigid Body Rotation
Rigid body transformations allow only rotation and translation
Rotation matrices form SO(3)
Special orthogonal group
(Distance preserving)
(No reflection)

50. Rigid Body Rotation R is normalized R is orthogonal
The squares of the elements in any row or column sum to 1
R is orthogonal
The dot product of any pair of rows or any pair columns is 0
The rows (columns) of R correspond to the vectors of the principle axes of the rotated coordinate frame

51. 3D Rotation About Arbitrary Axis

52. 3D Rotation About Arbitrary Axis
Translation : rotation axis passes through the origin
2. Make the rotation axis on the z-axis
3. Do rotation
4. Rotation & translation

53. 3D Rotation About Arbitrary Axis
Rotate u onto the z-axis

54. 3D Rotation About Arbitrary Axis
Rotate u onto the z-axis
u’: Project u onto the yz-plane to compute angle a
u’’: Rotate u about the x-axis by angle a
Rotate u’’ onto the z-asis

55. 3D Rotation About Arbitrary Axis
Rotate u’ about the x-axis onto the z-axis
Let u=(a,b,c) and thus u’=(0,b,c)
Let uz=(0,0,1)

56. 3D Rotation About Arbitrary Axis
Rotate u’ about the x-axis onto the z-axis
Since we know both cos a and sin a, the rotation matrix can be obtained
Or, we can compute the signed angle a
Do not use acos() since its domain is limited to [-1,1]

57. Euler angles
Arbitrary rotation can be represented by three rotation along x,y,z axis

58. Gimble
Hardware implementation of Euler angles
Aircraft, Camera

59. Euler Angles Rotation about three orthogonal axes 12 combinations
XYZ, XYX, XZY, XZX
YZX, YZY, YXZ, YXY
ZXY, ZXZ, ZYX, ZYZ
Gimble lock
Coincidence of inner most and outmost gimbles’ rotation axes
Loss of degree of freedom

60. Euler Angles Euler angles are ambiguous
Two different Euler angles can represent the same orientation
This ambiguity brings unexpected results of animation where frames are generated by interpolation.

61. Taxonomy of Transformations
Linear transformations
3x3 matrix
Rotation + scaling + shear
Rigid transformations
SO(3) for rotation
3D vector for translation
Affine transformation
3x3 matrix + 3D vector or 4x4 homogenous matrix
Linear transformation + translation
Projective transformation
4x4 matrix
Affine transformation + perspective projection

62. Taxonomy of Transformations
Rigid
Affine
Projective

63. Composite Transformations
Composite 2D Translation

64. Composite Transformations
Composite 2D Scaling

65. Composite Transformations
Composite 2D Rotation