1 Homogeneous Coordinates and Transformation

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

3 Line in R 2 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–2uu T ) Rotation about origin Shear

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

7 Definitions Equivalence relation ~ on the set S = R 3 \ {(0,0,0)} Ex: Show that this relation is reflexive, symmetric, and transitive Equivalence classes of the relation ~ Homogeneous coordinates Projective plane P 2 : the set of all equivalence classes An equivalence class is referred to as a point in the projective plane.

8 Definitions Choose a representative (u/w, v/w, 1) 1-1 correspondence with Cartesian plane Points on P 2 : I. [(u,v,w)] with w  0 II. [(u,v,w)] with w = 0 Corresponds to points-at-infinity, each with a specific direction Points on P 2 : the plane R 2 plus all the points at infinity

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

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

11 Visualization Line model [and spherical model]

Visualization 12

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

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

15 Two parallel linesDefining a line with a point at infinity

16 Plane in Cartesian Space Extend to P 3 and R 3

17 Intersection of Three Planes

18 Line in R 3 (Pl ü cker Coordinate) Line in parametric form Define Plucker coordinate of the line (q, q 0 ) p q q0q0

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

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: Point u becomes v = Hu Plane P’ becomes P H -1 Reason: After transformation H Note if P is written as a column vector, the formula becomes P ’ = H -T P

23

24 Transformation v = Hu

25 From Opengl-1.ppt

26 Vectors and Points are Different! 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)ref glVertex glNormal

27 v’=Mv won’t work

28 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 =M T, hence (M T ) T = M: rotate the vector as before For scaling:

29 Hence This is known as the normal matrix (ref)ref