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

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

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

4. Mathieu Brévilliers, Laboratoire MIA, UHA Delaunay triangulations

5. Mathieu Brévilliers, Laboratoire MIA, UHA Duality

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

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

8. Mathieu Brévilliers, Laboratoire MIA, UHA Elementary partitions

9. Mathieu Brévilliers, Laboratoire MIA, UHA Elementary partitions

10. Mathieu Brévilliers, Laboratoire MIA, UHA Elementary partitions

11. Mathieu Brévilliers, Laboratoire MIA, UHA Elementary partitions

12. Mathieu Brévilliers, Laboratoire MIA, UHA Elementary partitions

13. Mathieu Brévilliers, Laboratoire MIA, UHA Elementary partitions

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

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

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

17. Mathieu Brévilliers, Laboratoire MIA, UHA Topology

18. Mathieu Brévilliers, Laboratoire MIA, UHA Topology

19. Mathieu Brévilliers, Laboratoire MIA, UHA Topology

20. Mathieu Brévilliers, Laboratoire MIA, UHA Topology

21. Mathieu Brévilliers, Laboratoire MIA, UHA Topology

22. Mathieu Brévilliers, Laboratoire MIA, UHA Topology

23. Mathieu Brévilliers, Laboratoire MIA, UHA Topology

24. Mathieu Brévilliers, Laboratoire MIA, UHA Topology

25. Mathieu Brévilliers, Laboratoire MIA, UHA Topology

26. Mathieu Brévilliers, Laboratoire MIA, UHA Topology

27. Mathieu Brévilliers, Laboratoire MIA, UHA 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

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

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

30. Mathieu Brévilliers, Laboratoire MIA, UHA Duality

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

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

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

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

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

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

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

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