1. Reducing the degree of Apollonius Diagram Predicates David Millman Advisors: Sylvain Pion and Christophe Delage July 6th 2006

2. Background o Voronoi Diagram o Apollonius Diagram o Power Diagram o Incremental Algorithm o Invert and project

3. Voronoi Diagram

4. Apollonius Diagram Weighed point or Site s i is defined by p i, the point and w i the weight

5. Apollonius Graph Dual of Apollonius Diagram Vertex is a Site Edge two sites whose AW-Voronoi Cell share a face

6. Power Diagram

7. Incremental Algorithm The Three basic steps 1. Locate nearest neighbor 2. Check if new site is trivial 3. Update Vertex Conflict Edge Conflict

8. Vertex Conflict

9. Edge Conflict

10. Invert and Project

12. Invert and Project Convex Hull

13. Predicates Vertex Conflict, Edge Conflict Sub Predicates Orientation, Radical Side, Radical Intersection, Power Test, Order On a Line

14. Orientation

15. Radical Side

16. Radical Intersection

17. Vertex Conflict (revisited) There are 6 cases, and we have vertex conflict when: O-, RI-, RS{+|-} O+, RI+, RS+ O-, RI+, RS+

18. Vertex Conflict (degeneracies) When Orientation is 0 When Radical Intersection is 0

19. Pertubations Consistent ordering 1. Max Radius 2. Lexigraphically

20. Edge Conflict (revisited) We are looking for the case of q breaking 2 into multiple sections, when inverted and projection onto the unit sphere.

21. Numerical Results 66% speedup for vertex conflict 4 finite sites 39% speedup for vertex conflict 3 finite and 1 infinite site 10-20% lessfilter failures in nearly degenerate cases

22. Further Work Optimize Edge Conflict Optimizations of the Incremental Algorithm Degeneracies in 3D Reduction of Exact Computations

23. Reducing the degree of Apollonius Diagram Predicates Merci!!!