1 / 30

Homogeneous Coordinates and Transformation

Homogeneous Coordinates and Transformation. (x,y). n. Distance to origin. For any two points on the line:.  n  line. (projection along n ). Line in R 2. General line equation. Normalize:. Line in R 2. Parametric equation of a line. Corresponding implicit form:. Implicitize:.

nferraro
Download Presentation

Homogeneous Coordinates and Transformation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Homogeneous Coordinates and Transformation

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

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

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

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

  16. Extend to P3 and R3 Plane in Cartesian Space

  17. Intersection of Three Planes

  18. q0 p q Line in R3(Plücker Coordinate) Line in parametric form 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

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

  24. v = Hu Transformation

  25. From Opengl-1.ppt

  26. 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 Vectors and Points are Different! glNormal glVertex (ref)

  27. v’=Mv won’t work

  28. 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: On (M-1)T

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

More Related