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

2. Pseudo-Triangle 3 corners non-corners

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 detection P.K.Agarwal, J.Basch, L.J.Guibas, J.Hershberger, L.Zhang. Deformable free space tilings for kinetic collision detection. 2001 D.Kirkpatrick, J.Snoeyink, B.Speckmann. Kinetic collision detection for simple polygons. 2002 D.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. 2000 G.Rote, F.Santos, I.Streinu. Expansive motions and the polytope of pointed pseudo-triangulations. 2001 R.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. 1999 B.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. Projectivity projective edges of surface project vertically to edges of graph regular surface is in convex position

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

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

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

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. Projectivity not projective edges

24. Projectivity A pseudo-triangulation is stable if no subset of pending points can be eliminated with their incident edges s.t. (1) a valid pseudo-triangulation remains (2) status of each point is unchanged

25. Stability

26. remove both pending points: no valid pseudo-triangulation

27. Stability remove right pending point: no valid pseudo-triangulation

28. Stability remove left pending point: status changes

29. Stability  stable

30. Stability  not stable

31. 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.

32. Surface Flips

33. triangulations: tetrahedral flips, Lawson flips edge-exchangingpoint removing/inserting

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

35. Surface Flips flip reflex edge

36. Surface Flips convexifying flip

37. Surface Flips new flip type in pseudo-triangulations edge-removing/inserting 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

38. Surface Flips flip reflex edge

39. Surface Flips planarizing flip

40. 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

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

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

43. 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.

44. Optimality Theorem special cases: O(n 2 ) surface flips - (P,S) convex: lower convex hull - (P,S) polygon without interior points

45. Optimality Theorem initial surface flip

46. Optimality Theorem flip 1: convexifying flip

47. Optimality Theorem flip 2: planarizing flip

48. Optimality Theorem flip 3: planarizing flip

49. Optimality Theorem flip 4: convexifyingoptimum

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

51. Optimality Theorem initial triangulation

52. Optimality Theorem lifted surface

53. Optimality Theorem lifted surface flip

54. Optimality Theorem flip 1: planarizing flip

55. Optimality Theorem flip 2: planarizing flip

56. Optimality Theorem flip 3: planarizing remove edges

57. Optimality Theorem optimum

58. Optimality Theorem every triangulation surface can be flipped to regularity with surface flips generalization of situation for Delaunay triangulation (convex heights)

59. admissible planar straight-line graph: each component is connected to the boundary Constrained Regularity  collection of polygons with interior points Optimality Theorem: f * piecewise linear, but not continuous in general

60. Polytope Representation convex polytope: all regular pseudo-triangulations constrained by an admissible planar straight-line graph generalization of associahedron (secondary polytope) for regular triangulations

61. Spatial Embedding of Pseudo-Triangulations Thank you!