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Overlay Stitch Meshing

Overlay Stitch Meshing. A competitive algorithm for no-large-angle triangulation. Don Sheehy Joint work with Gary Miller and Todd Phillips To appear at ICALP 2007. The Problem. Input: A Planar Straight Line Graph. The Problem. Input: A Planar Straight Line Graph.

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Overlay Stitch Meshing

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  1. Overlay Stitch Meshing Don Sheehy Overlay Stitch Meshing

  2. A competitive algorithm for no-large-angle triangulation Don Sheehy Joint work with Gary Miller and Todd Phillips To appear at ICALP 2007 Don Sheehy Overlay Stitch Meshing

  3. The Problem Input: A Planar Straight Line Graph Don Sheehy Overlay Stitch Meshing

  4. The Problem Input: A Planar Straight Line Graph Output: A Conforming Triangulation Don Sheehy Overlay Stitch Meshing

  5. The Problem Input: A Planar Straight Line Graph Output: A Conforming Triangulation Don Sheehy Overlay Stitch Meshing

  6. Why would you want to do that? Don Sheehy Overlay Stitch Meshing

  7. Why would you want to do that? Don Sheehy Overlay Stitch Meshing

  8. Why would you want to do that? Don Sheehy Overlay Stitch Meshing

  9. Why would you want to do that? Don Sheehy Overlay Stitch Meshing

  10. Why would you want to do that? Don Sheehy Overlay Stitch Meshing

  11. Why would you want to do that? Don Sheehy Overlay Stitch Meshing

  12. Don Sheehy Overlay Stitch Meshing

  13. Don Sheehy Overlay Stitch Meshing

  14. Don Sheehy Overlay Stitch Meshing

  15. Don Sheehy Overlay Stitch Meshing

  16. Don Sheehy Overlay Stitch Meshing

  17. Don Sheehy Overlay Stitch Meshing

  18. What went wrong? Don Sheehy Overlay Stitch Meshing

  19. ? ? What if you don’t know the function? Don Sheehy Overlay Stitch Meshing

  20. 2 Definitions of Quality 1. No Large Angles [Babuska, Aziz 1976] Don Sheehy Overlay Stitch Meshing

  21. 2 Definitions of Quality 1. No Large Angles [Babuska, Aziz 1976] 2. No Small Angles Don Sheehy Overlay Stitch Meshing

  22. No Small Angles Don Sheehy Overlay Stitch Meshing

  23. No Small Angles You may have heard of these before. Delaunay Refinement Sparse Voronoi Refinement Quadtree Don Sheehy Overlay Stitch Meshing

  24. Paying for the spread Don Sheehy Overlay Stitch Meshing

  25. Paying for the spread Spread = L/s L s Don Sheehy Overlay Stitch Meshing

  26. Paying for the spread Optimal No-Large-Angle Triangulation Don Sheehy Overlay Stitch Meshing

  27. Paying for the spread What if we don’t allow small angles? Don Sheehy Overlay Stitch Meshing

  28. Paying for the spread What if we don’t allow small angles? Don Sheehy Overlay Stitch Meshing

  29. Paying for the spread What if we don’t allow small angles? O(L/s) triangles! Don Sheehy Overlay Stitch Meshing

  30. Paying for the spread What if we don’t allow small angles? Fact: For inputs with NO edges, no-small-angle meshing algorithms produce output with O(n log L/s) size and angles between 30o and 120o O(L/s) triangles! Don Sheehy Overlay Stitch Meshing

  31. What to do? Don Sheehy Overlay Stitch Meshing

  32. What to do? Small input angles can force even smaller ouput angles. [Shewchuk ’02] Don Sheehy Overlay Stitch Meshing

  33. No Large Angles Don Sheehy Overlay Stitch Meshing

  34. Polygons with Holes [Bern, Mitchell, Ruppert 95] – - All triangles are nonobtuse. - Output has O(n) triangles. Don Sheehy Overlay Stitch Meshing

  35. Polygons with Holes [Bern, Mitchell, Ruppert 95] – - All triangles are nonobtuse. - Output has O(n) triangles. Does not work for arbitrary PSLGs Don Sheehy Overlay Stitch Meshing

  36. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing

  37. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing

  38. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing

  39. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing

  40. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing

  41. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing

  42. Propagating Paths Don Sheehy Overlay Stitch Meshing

  43. Propagating Paths Don Sheehy Overlay Stitch Meshing

  44. Propagating Paths Don Sheehy Overlay Stitch Meshing

  45. Propagating Paths Don Sheehy Overlay Stitch Meshing

  46. Propagating Paths Don Sheehy Overlay Stitch Meshing

  47. Propagating Paths First introduced by Mitchell [93] Don Sheehy Overlay Stitch Meshing

  48. Propagating Paths First introduced by Mitchell [93] Later Improved by Tan [96] Don Sheehy Overlay Stitch Meshing

  49. Propagating Paths First introduced by Mitchell [93] Later Improved by Tan [96] Worst Case Optimal Size O(n2) Angle bounds: 132o Don Sheehy Overlay Stitch Meshing

  50. The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing

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