Spatial Embedding of Pseudo-Triangulations - PowerPoint PPT Presentation

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Spatial Embedding of Pseudo-Triangulations

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  1. Spatial Embedding of Pseudo-Triangulations Oswin Aichholzer Institute for Software Technology Graz University of Technology Graz, Austria Franz Aurenhammer Hannes Krasser Institute for Theoretical Computer Science Graz University of Technology Graz, Austria Peter Braß Institut für Informatik Freie Universität Berlin Berlin, Germany supported by Apart, FWF, DFG

  2. non-corners 3 corners Pseudo-Triangle

  3. Pseudo-Triangulation

  4. Applications ray shooting B.Chazelle, H.Edelsbrunner, M.Grigni, L.J.Guibas, J.Hershberger, M.Sharir, J.Snoeyink. Ray shooting in polygons using geodesic triangulations. 1994 M.T.Goodrich, R.Tamassia. Dynamic ray shooting and shortest paths in planar subdivisions via balanced geodesic triangulations. 1997 visibility M.Pocchiola, G.Vegter. Minimal tangent visibility graphs. 1996 M.Pocchiola, G.Vegter. Topologically sweeping visibility complexes via pseudo-triangulations. 1996 kinetic collision detectionP.K.Agarwal, J.Basch, L.J.Guibas, J.Hershberger, L.Zhang. Deformable free space tilings for kinetic collision detection. 2001D.Kirkpatrick, J.Snoeyink, B.Speckmann. Kinetic collision detection for simple polygons. 2002D.Kirkpatrick, B.Speckmann. Kinetic maintenance of context-sensitive hierarchical representations for disjoint simple polygons. 2002

  5. Applications rigidity I.Streinu. A combinatorial approach to planar non-colliding robot arm motion planning. 2000G.Rote, F.Santos, I.Streinu. Expansive motions and the polytope of pointed pseudo-triangulations. 2001R.Haas, F.Santos, B.Servatius, D.Souvaine, I.Streinu, W.Whiteley. Planar minimally rigid graphs have pseudo-triangular embeddings. 2002 guarding M.Pocchiola, G.Vegter. On polygon covers. 1999B.Speckmann, C.D.Toth. Allocating vertex Pi-guards in simple polygons via pseudo-triangulations. 2002

  6. Overview • pseudo-triangulation surfaces • new flip type • locally convex functions

  7. Triangulations set of points in the plane assume general position

  8. Triangulations triangulation in the plane

  9. Triangulations assign heights to each point

  10. Triangulations lift points to assigned heights

  11. Triangulations spatial surface

  12. Triangulations spatial surface

  13. regular surface is in convex position Projectivity projective edges of surface project vertically to edges of graph

  14. Pseudo-Triangulations more general: polygon with interior points set of points in the plane partition points rigid points corner in all incident pseudo-triangles pending points non-corner in one incident pseudo-triangle

  15. Surface Theorem Theorem. Let (P,S) be a polygon with interior points, and let PT be any pseudo-triangulation thereof. Let h be a vector assigning a height to each rigid vertex of PT. For each choice of h, there exists a unique polyhedral surface F above P, that respects h and whose edges project vertically to (a subset of) the edges of PT.

  16. Surface Theorem pseudo-triangulation in the plane

  17. Surface Theorem surface

  18. Surface Theorem surface surface

  19. linear system: Surface Theorem sketch of proof: rigid points: fixed height pending points: co-planar with 3 corners

  20. Surface Theorem rigid points pending points

  21. Surface Theorem Theorem. Let (P,S) be a polygon with interior points, and let PT be any pseudo-triangulation thereof. Let h be a vector assigning a height to each rigid vertex of PT. For each choice of h, there exists a unique polyhedral surface F above P, that respects h and whose edges project vertically to (a subset of) the edges of PT.

  22. Surface Theorem Theorem. Let (P,S) be a polygon with interior points, and let PT be any pseudo-triangulation thereof. Let h be a vector assigning a height to each rigid vertex of PT. For each choice of h, there exists a unique polyhedral surface F above P, that respects h and whose edges project vertically to (a subset of) the edges of PT.

  23. not projective edges Projectivity

  24. Projectivity • A pseudo-triangulation is stable • if no subset of pending points can be • eliminated with their incident edges s.t. • a valid pseudo-triangulation remains • (2) status of each point is unchanged

  25. Projectivity Theorem. A pseudo-triangulation PT of (P,S) is projective only if PT is stable. If PT is stable then the point set S can be perturbed (by some arbitrarily small ε) such that PT becomes projective.

  26. Surface Flips

  27. Surface Flips triangulations: tetrahedral flips, Lawson flips edge-exchanging point removing/inserting

  28. Surface Flips flips in pseudo-triangulations edge-exchanging, geodesics

  29. Surface Flips flip reflex edge

  30. Surface Flips convexifying flip

  31. Surface Flips new flip type in pseudo-triangulations independently introduced by D.Orden, F.Santos. The polyhedron of non-crossing graphs on a planar point set. 2002 also in O. Aichholzer, F. Aurenhammer, and H. Krasser. Adapting (pseudo-) triangulations with a near-linear number of edge flips. WADS 2003 edge-removing/inserting

  32. Surface Flips flip reflex edge

  33. Surface Flips planarizing flip

  34. Locally Convex Functions P … polygon in the plane f … real-valued function with domain P locally convex function: convex on each line segment interior to P

  35. Locally Convex Functions optimization problem: (P,S) … polygon with interior points h … heights for points in S f * … maximal locally convex function with f*(vi) ≤ hi for each viS

  36. Locally Convex Functions properties of f *: - unique and piecewise linear - corresponding surface F * projects to a pseudo-triangulation of (P,S‘), S‘S

  37. Optimality Theorem Theorem: Let F*(T,h) be a surface obtained from F(T,h) by applying convexifying and planarizing surface flips (in any order) as long as reflex edges do exist. Then F*(T,h)=F*, for any choice of the initial triangulation T. The optimum F* is reached after a finite number of surface flips.

  38. flip Optimality Theorem initial surface

  39. Optimality Theorem flip flip 1: convexifying

  40. Optimality Theorem flip flip 2: planarizing

  41. Optimality Theorem flip flip 3: planarizing

  42. Optimality Theorem flip 4: convexifying optimum

  43. reflex convex Optimality Theorem tetrahedral flips are not sufficient to reach optimality 0 1 1 1 0 0

  44. Optimality Theorem initial triangulation

  45. Optimality Theorem lifted surface

  46. Optimality Theorem flip lifted surface

  47. Optimality Theorem flip flip 1: planarizing

  48. Optimality Theorem flip flip 2: planarizing

  49. Optimality Theorem remove edges flip 3: planarizing

  50. Optimality Theorem optimum