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Elementary Partitions of Line Segments in the Plane

Elementary Partitions of Line Segments in the Plane. Point set triangulations. Point set triangulations. Delaunay triangulations. Duality. Legal edge property. Legal edge property. The Delaunay triangulation is the unique triangulation without illegal edge.

joan-blackburn
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Elementary Partitions of Line Segments in the Plane

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  1. Elementary Partitions of Line Segments in the Plane Mathieu Brévilliers, Laboratoire MIA, UHA

  2. Point set triangulations Mathieu Brévilliers, Laboratoire MIA, UHA

  3. Point set triangulations Mathieu Brévilliers, Laboratoire MIA, UHA

  4. Delaunay triangulations Mathieu Brévilliers, Laboratoire MIA, UHA

  5. Duality Mathieu Brévilliers, Laboratoire MIA, UHA

  6. Legal edge property Mathieu Brévilliers, Laboratoire MIA, UHA

  7. Legal edge property • The Delaunay triangulation is the unique triangulation without illegal edge. O(n) checker for Delaunay triangulations Mathieu Brévilliers, Laboratoire MIA, UHA

  8. Elementary partitions Mathieu Brévilliers, Laboratoire MIA, UHA

  9. Elementary partitions Mathieu Brévilliers, Laboratoire MIA, UHA

  10. Elementary partitions Mathieu Brévilliers, Laboratoire MIA, UHA

  11. Elementary partitions Mathieu Brévilliers, Laboratoire MIA, UHA

  12. Elementary partitions Mathieu Brévilliers, Laboratoire MIA, UHA

  13. Elementary partitions Mathieu Brévilliers, Laboratoire MIA, UHA

  14. Shape of edges Mathieu Brévilliers, Laboratoire MIA, UHA

  15. Shape of edges Mathieu Brévilliers, Laboratoire MIA, UHA

  16. Shape of edges Mathieu Brévilliers, Laboratoire MIA, UHA

  17. Topology Mathieu Brévilliers, Laboratoire MIA, UHA

  18. Topology Mathieu Brévilliers, Laboratoire MIA, UHA

  19. Topology Mathieu Brévilliers, Laboratoire MIA, UHA

  20. Topology Mathieu Brévilliers, Laboratoire MIA, UHA

  21. Topology Mathieu Brévilliers, Laboratoire MIA, UHA

  22. Topology Mathieu Brévilliers, Laboratoire MIA, UHA

  23. Topology Mathieu Brévilliers, Laboratoire MIA, UHA

  24. Topology Mathieu Brévilliers, Laboratoire MIA, UHA

  25. Topology Mathieu Brévilliers, Laboratoire MIA, UHA

  26. Topology Mathieu Brévilliers, Laboratoire MIA, UHA

  27. Numbers of edges and faces • 3n – n’ – 3 edges • 2n – n’ – 2 faces • n : number of sites • n’ : number of sides of the CH that are not sites Mathieu Brévilliers, Laboratoire MIA, UHA

  28. Elementary Delaunay partition Mathieu Brévilliers, Laboratoire MIA, UHA

  29. Elementary Delaunay partition Mathieu Brévilliers, Laboratoire MIA, UHA

  30. Duality Mathieu Brévilliers, Laboratoire MIA, UHA

  31. Legal edge property Mathieu Brévilliers, Laboratoire MIA, UHA

  32. Legal edge property Legal Illegal Mathieu Brévilliers, Laboratoire MIA, UHA

  33. Legal edge property • The elementary Delaunay partition is the unique elementary partition without illegal edge. Mathieu Brévilliers, Laboratoire MIA, UHA

  34. Checker for Delaunay topology Topology Geometry with faces in Delaunay positions Is it an elementary partition ? Mathieu Brévilliers, Laboratoire MIA, UHA

  35. d a c b a d b c Checker for Delaunay topology   Mathieu Brévilliers, Laboratoire MIA, UHA

  36. Checker for Delaunay topology 1. For each edge, the test runs in constant time 2. O(n) edges in an elementary partition Linear algorithm Mathieu Brévilliers, Laboratoire MIA, UHA

  37. Future works • Flip algorithm • Equiangularity for elementary Delaunay partitions • More general set of sites • Higher dimensions Mathieu Brévilliers, Laboratoire MIA, UHA

  38. Thank you! Mathieu Brévilliers, Laboratoire MIA, UHA

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