1. Homogeneous Coordinates and Transformation

2. Line in R2 General line equation n Normalize: Distance to origin
(x,y)
General line equation
Normalize:
For any two points on the line:
 n  line
Distance to origin
(projection along n)

3. Line in R2 Parametric equation of a line Corresponding implicit form:
Implicitize:

4. Affine Transformation
Properties:
Collinearity (maps a line to a line)
Preserve ratio of distances (midpoint stays in the middle after transformation)

5. Common 2D Affine Transformations
Translation
Scaling
Reflection (Q = I–2uuT)
Rotation about origin
Shear

6. Homogeneous Coordinate
Motivation: to unify representations of affine map (esp. translation)

7. Definitions Equivalence relation ~ on the set S = R3 \ {(0,0,0)}
Ex: Show that this relation is reflexive, symmetric, and transitive
Equivalence classes of the relation ~
Homogeneous coordinates
Projective plane P2: the set of all equivalence classes
Reflexive: (a,a)  R  a  A
Symmetric: if (b,a)R whenever (a,b)R
Transitive: whenever (a,b)R and (b,c)R then (a,c) R
An equivalence class is referred to as a point in the projective plane.

8. Definitions Points on P2: I. [(u,v,w)] with w 0
Choose a representative (u/w, v/w, 1)
1-1 correspondence with Cartesian plane
II. [(u,v,w)] with w = 0
Corresponds to points-at-infinity, each with a specific direction
Points on P2: the plane R2 plus all the points at infinity

9. Reach the same point (at ), from any starting point
Points at Infinity
(x,y,0)
Points at infinity: (x,y,0)
Reach the same point (at ), from any starting point

10. Parallel Lines Intersect at Infinity
(-2,1,0)

11. Visualization
Line model [and spherical model]

12. Visualization

13. Line in Cartesian Space
(or any multiple of it)
(or any multiple of it)

14. Examples (cases in R2) The line passes through (3,1) and (-4,5)
Intersection of

15. Two parallel lines
Defining a line with a point at infinity

16. Plane in Cartesian Space
Extend to P3 and R3
Plane in Cartesian Space

17. Intersection of Three Planes

18. Line in R3 (Plücker Coordinate)
Line in parametric form p q q0
Define
Plucker coordinate of the line (q, q0)

19. Space Transformation Translation Scaling
Rotation about coordinate axes
Rotation about arbitrary line
Reflection about arbitrary plane (Q=I–2uuT)

20. Transformed Equations
If transformation T is applied to geometry (line/plane), what’s the transformed equation?
Apply T to homogenous line/plane equation?! NOT !!
Answers:
See handout p.3 (convert to parametric form; transform the points; then to implicit equation)
More detailed version: see “homogeneous-transformation.ppt” from R. Paul (next page)
Also related to the normal matrix in OpenGL.

21. From Richard Paul Ch.1

22. Summary Point u on a plane: After transformation H
Point u becomes v = Hu
Plane P’ becomes PH-1
Reason:
Note if P is written as a column vector, the formula becomes P’ = H-TP

24. Transformed Quadrics Point u on a quadric: After transformation H
Point u becomes v = Hu
Quadric Q becomes H-TQH-1
Reason:

25. v = Hu
Transformation

26. From Opengl-1.ppt

27. Vectors and Points are Different!
glVertex
glNormal
Point
Homogenenous coordinate
p = [x y z 1]
M: affine transform (translate, rotate, scaling, reflect, …)
p’= M p
Vector
Homogeneous coordinate
v = [x y z 0]
Affine transform (applicable when M is invertible (not full rank; projection to 2D is not)
v’= (M-1)T v
(ref)

28. v’=Mv won’t work

29. On (M-1)T
The w (homogeneous coord) of vectors are 0; hence, the translation part (31 vector) plays no role
For rotation, M-1=MT, hence (MT)T = M: rotate the vector as before
For scaling:

30. Hence
This is known as the normal matrix (ref)