1. The Voronoi diagram of convex objects in the plane Menelaos Karavelas & Mariette Yvinec Dagsthul Workshop, march 2003

2. A Voronoi diagram : a set of site + a distance

3. The distance

4. The set of sites 1.Pseudo-circles sets of smooth convex objects convex objects smooth boundaries at most two intersection points between the boundaries of two objects 2.Pseudo-circles sets of piecewise smooth convex objects 3.General sets of convex objects Voronoi diagram restricted outside the complement of the union

5. Pseudo-circles sets of smooth convex objects

6. Pseudo-circles sets of piecewise smooth convex objects

7. Non pseudo circles set

8. Previous works Concrete and abstract Voronoi diagrams Klein 89 Randomized incremental construction on abstract Voronoi diagrams Klein, Mehlhorn & Meiser 93 The Voronoi diagram of curved objects. Alt & Schwarzkopf 95 Dynamic additively weighted Voronoi diagrams in 2D Karavelas & Yvinec 02

9. The case of pseudo-circles sets of smooth convex objects Th1 : The bisector of two sites is –either empty –or a single curve homeomorphic to ]0,1[ Th2 : The cell of each site is simply connected

10. The cell of an object Non empty cells Empty cells Hidden object Ai : any maximal disk in Ai is included in some other Aj

11. Th1 : The bisector of two sites is –either empty –or a single curve homeomorphic to ]0,1[ bisector empty

12. consider the function

13. Th2 : Voronoi cells are simply connected 1)

14. 2)

15. The Algorithm Randomized incremental The basic data structures The 1-skeleton of the Voronoi diagram or its dual graph The covering graph: to keep track of hidden sites

16. The conflict region : when inserting new site A part of the Voronoi 1-skeleton where bitangent circles are either internal and included in A or external and intersecting A

17. Insertion of a new site Find a first conflict or a covering of the new site Find the while conflict region and repair the Voronoi diagram update the covering graph

18. Nearest neighbor query : the Voronoi hierarchy Construction Level 0 : the whole diagram Level k : insert each sites in level k-1 with propability  Query At each level, the visited sites have decreasing distances to the query point Expected number of sites visited : O(1/  )

19. + a tree for each cell To avoid checking all the neighbors of A to find one closer to q, the normals through the Voronoi vertices of the cell of A are stored in a bb-tree Time spend in each cell O( log n) nn-query time

20. Find the first conflict or detect a hidden site Disjoint sites –issue a nn query for a point p of A –at least one edge of cell(nn(p)) conflicts A Intersecting sites –issue a nn query for any point p of ma(A) –if M(p)  nn(p), at least one edge of cell(nn(p)) conflicts with A –if M(p)  nn(p) prune ma(A) and iterate

21. Removal of object A Update the Voronoi diagram –insertion the neighbors of A in an annex Voronoi diagram –copy back in the main diagram the filling of cell(A) Remove A from covering graph Reinsert objects hidden by A

22. Voronoi diagram for pseudo-circles set Expected complexity ObjectsdisjointsNo hiddenHidden Insertion Deletion

23. Pseudo-circles sets of piecewise smooth objects add point site at the vertices perturb the distance

24. General convex objects

25. Further work work out the predicates and implement the algorithm for ellipses extend to pseudo-circles set of non convex objects Voronoi diagram for convex objects in 3d