Problems in curves and surfaces

1. Problems in curves and surfaces M. Ramanathan Problems in curves and surfaces

2. Simple problems • Given a point p and a parametric curve C(t), find the minimum distance between p and C(t) C(t) <p – C(t), C’(t)> = 0 Constraint equation p Problems in curves and surfaces

3. Point-curve tangents Given a point p and a parametric curve C(t), find the tangents from p to C(t) Problems in curves and surfaces

4. Common tangent lines Problems in curves and surfaces

5. The IRIT Modeling Environment • www.cs.technion.ac.il/~irit • More like a kernal not a software – code can be downloaded from the same webpage. • Add your own functions and compile with them (written in C language) • User’s manual as well as programming manual is available Problems in curves and surfaces

6. 4 5 3 6 2 1 0 Convex hull of a point set • Given a set of pins on a pinboard • And a rubber band around them • How does the rubber band look when it snaps tight? • A CH is a convex polygon - non-intersecting polygon whose internal angles are all convex (i.e., less than π) Problems in curves and surfaces

7. Bi-Tangents and Convex hull Problems in curves and surfaces

8. CH of closed surfaces Problems in curves and surfaces

9. CH of closed surfaces Problems in curves and surfaces

10. Minimum enclosing circle • smallest circle that completely contains a set of points Problems in curves and surfaces

11. Minimum enclosing circle – two curves Problems in curves and surfaces

12. Minimum enclosing circle – three curves Problems in curves and surfaces

13. MEC of a set of closed curves Problems in curves and surfaces

14. Kernel problem • Given a freeform curve/surface, find a point from which the entire curve/surface is visible. Problems in curves and surfaces

15. Kernel problem (contd.) Solve Problems in curves and surfaces

16. Kernel problem in surfaces Problems in curves and surfaces

17. Duality • duality refers to geometric transformations that replace points by lines and lines by points while preserving incidence properties among the transformed objects. The relations of incidence are those such as 'lies on' between points and lines (as in 'point P lies on line L') Problems in curves and surfaces

18. Point-Line Duality Problems in curves and surfaces

19. Common tangents Problems in curves and surfaces

20. Voronoi Cell (Points) • Given a set of points {P1, P2, … , Pn}, the Voronoi cell of point P1 is the set of all points closer to P1 than to any other point. Problems in curves and surfaces

21. Skeleton – Voronoi diagram The Voronoi diagram is the union of the Voronoi cells of all the free-form curves. Problems in curves and surfaces

22. Voronoi diagram (illustration) P3 B(P2,P3) B(P1,P3) P1 P2 P2 P1 Remember that VD is not defined for just points but for any set e.g. curves, surfaces etc. Moreover, the definition is applicable for any dimension. B(P1,P2) B(P1,P2) Problems in curves and surfaces

23. Skeleton – Medial Axis The medial axis (MA), or skeleton of the set D, is defined as the locus of points inside which lie at the centers of all closed discs (or balls in 3-D) which are maximal in D. Problems in curves and surfaces

24. Skeletons – medial axis Problems in curves and surfaces

25. Definition (Voronoi Cell) C2(r2) • Given - C0(t), C1(r1), ... , Cn(rn) - disjoint rational planar closed regular C1 free-form curves. • The Voronoi cell of a curve C0(t) is the set of all points closer to C0(t) than to Cj(rj), for all j > 0. C1(r1) C0(t) C3(r3) C4(r4) Problems in curves and surfaces

26. Definition (Voronoi cell (Contd.)) C2(r2) • Boundary of the Voronoi cell. • Voronoi cell consists of points that are equidistant and minimal from two differentcurves. C1(r1) C0(t) C3(r3) C3(r3) C0(t), C4(r4) C0(t), C4(r4) Problems in curves and surfaces

27. Definition (Voronoi cell (Contd.)) “The Voronoi cell consists of points that are equidistant and minimal from two differentcurves.” • The above definition excludes non-minimal-distance bisector points. • This definition excludes self-Voronoi edges. r3 r4 r2 r C1(r) r1 q p t C0(t) Problems in curves and surfaces

28. Definition (Voronoi diagram) The Voronoi diagram is the union of the Voronoi cells of all the free-form curves. C0(t) Problems in curves and surfaces

29. Skeleton-Bisector relation Problems in curves and surfaces

30. Bisector for simple curves Problems in curves and surfaces

31. Bisector for simple curves (contd) Problems in curves and surfaces

32. Point-curve bisector Problems in curves and surfaces

33. C1(r) RR LR C0(t) LL RL Curve-curve bisector Problems in curves and surfaces

34. Euclidean space C1(r) C0(t) Splitting into monotone pieces Limiting constraints Lower envelope algorithm Outline of the algorithm tr-space Implicit bisector function Problems in curves and surfaces

35. P(t,r) - q The implicit bisector function • Given two regular C1 parametric curves C0(t) and C1(r), one can get a rational expression for the two normals’ intersection point: P(t,r) = (x(t,r), y(t,r)). • The implicit bisector function F3 is defined by: q Problems in curves and surfaces

36. F3(t,r) C1(r) t r C0(t) The untrimmed implicit bisector function Comment: Note we capture in the (finite) F3 the entire (infinite) bisector in R2. Problems in curves and surfaces

37. r t t Splitting the bisector, the zero-set of F3, into monotone pieces r Keyser et al., Efficient and exact manipulation of algebraic points and curves, CAD, 32 (11), 2000, pp 649--662. Problems in curves and surfaces

38. Constraints - orientation • Orientation Constraint – purge regions of the untrimmed bisector that do not lie on the proper side. • LL considers leftside of both curves as proper: Problems in curves and surfaces

39. C1(r) RR LR C0(t) LL RL The orientation constraints (Contd.) Problems in curves and surfaces

40. N1/κ1 C1(t1) P(t1, t2) C2(t2) The curvature constraints Curvature Constraint (CC) – purge away regions of the untrimmed bisector whose distance to its footpoints (i.e., the radius of the Voronoi disk) is larger than the radius of curvature (i.e., 1/κ) at the footpoint. Problems in curves and surfaces

41. Effect of the curvature constraint Problems in curves and surfaces

42. Application of curvature constraint Before After Problems in curves and surfaces

43. D D t D t (b) (a) t (c) Lower envelopes Problems in curves and surfaces

44. Lower envelope algorithm General Lower Envelope VC Lower Envelope Distance function D defined by Di(t, ri) = ||P(t, ri) - Ci(t) || • Standard Divide and Conquer algorithm. • Main needed functions are: • Identifying intersections of curves. • Comparing two curves at a given parameter (above/below). • Splitting a curve at a given parameter. • ||Di (t, ri)||2 = ||Dj(t, rj)||2 , F3(t, ri) = 0, F3(t, rj) = 0. • Compare ||Di(t, ri)||2 and ||Dj(t,rj)||2 at the parametric values. • Split F3(t, ri) = 0 at the tri-parameter. Problems in curves and surfaces

45. C0(t) C0(t) C0(t) C2(r2) C1(r1) C1(r1) C1(r1) Result I Problems in curves and surfaces

46. C2(r2) C0(t) C2(r2) C0(t) C1(r1) C1(r1) Result I (Contd.) Problems in curves and surfaces

47. C3(r3) C1(r1) C0(t) C1(r1) C2(r2) C0(t) C3(r3) C2(r2) C4(r4) Results II Problems in curves and surfaces

48. C2(r2) C2(r2) C3(r3) C0(t) C4(r4) C0(t) C1(r1) C1(r1) Results III Problems in curves and surfaces

49. C1(r1) C0(t) C2(r2) Results IV (For Non-Convex C0(t)) Voronoi cell is obtained by performing the lower envelope on both t and r parametric directions. C3(r3) C2(r2) C0(t) C1(r1) Problems in curves and surfaces

50. Bisectors in 3D Problems in curves and surfaces