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Review Computing the Delaunay Triangulation Analysis Conclusion Department of Computer Science, Yazd University Delaunay Triangulations-Part II A.Rahiminasab December 2014 1 / 52 Review Computing the Delaunay Triangulation Triangulations

  1. Review Computing the Delaunay Triangulation Triangulations of Planar Point Sets Analysis The Delaunay Triangulation Conclusion Review Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation; they tend to avoid skinny triangles. For a set of points on the same line there is no Delaunay triangulation (the notion of triangulation is degenerate for this case) For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique By considering circumscribed spheres, the notion of Delaunay triangulation extends to three and higher dimensions 13 / 52

  2. Review Computing the Delaunay Triangulation Triangulations of Planar Point Sets Analysis The Delaunay Triangulation Conclusion Review Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation; they tend to avoid skinny triangles. For a set of points on the same line there is no Delaunay triangulation (the notion of triangulation is degenerate for this case) For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique By considering circumscribed spheres, the notion of Delaunay triangulation extends to three and higher dimensions 13 / 52

  3. Review Computing the Delaunay Triangulation Triangulations of Planar Point Sets Analysis The Delaunay Triangulation Conclusion Review The triangulation is named after Boris Delaunay for his work on this topic from 1934. 14 / 52

  4. Review Computing the Delaunay Triangulation Triangulations of Planar Point Sets Analysis The Delaunay Triangulation Conclusion Review The voronoi diagram is named after Georgy F . Voronoi for his work on this topic. 15 / 52

  5. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Review 16 / 52

  6. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Review 17 / 52

  7. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Review 18 / 52

  8. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Usage Delaunay triangulations help in constructing various things: Euclidean Minimum Spanning Trees Approximations to the Euclidean Traveling Salesperson Problem , ... 19 / 52

  9. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Usage 20 / 52

  10. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Usage 21 / 52

  11. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Usage 22 / 52

  12. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Methods There are several ways to compute the Delaunay triangulation: By plane sweep By iterative flipping from any triangulation By conversion from the Voronoi diagram Go to diagram By randomized incremental approach √ 23 / 52

  13. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Methods There are several ways to compute the Delaunay triangulation: By plane sweep By iterative flipping from any triangulation By conversion from the Voronoi diagram Go to diagram By randomized incremental approach √ 23 / 52

  14. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P 0 24 / 52

  15. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  16. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  17. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  18. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  19. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  20. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  21. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  22. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  23. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  24. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  25. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  26. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  27. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  28. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  29. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  30. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  31. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 24 / 52 P -2

  32. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  33. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  34. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  35. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  36. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  37. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  38. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  39. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  40. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  41. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  42. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  43. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  44. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  45. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  46. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  47. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  48. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  49. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  50. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  51. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  52. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  53. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  54. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  55. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  56. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  57. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  58. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  59. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  60. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  61. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  62. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  63. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  64. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  65. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  66. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  67. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  68. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  69. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  70. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  71. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  72. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  73. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  74. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P -1 P 0 P -2 24 / 52

  75. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach P 0 24 / 52

  76. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Pseudocode Back 25 / 52

  77. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Pseudocode Back 25 / 52

  78. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Randomized incremental approach 26 / 52

  79. Usage Review Methods Computing the Delaunay Triangulation Description of the fourth method Analysis Algorithm Conclusion Point location structure Pseudocode 27 / 52

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