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Problems in curves and surfaces

Problems in curves and surfaces. M. Ramanathan. 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. Point-curve tangents.

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Problems in curves and surfaces

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  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

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