Reducing the degree of Apollonius Diagram Predicates David Millman Advisors: Sylvain Pion and Christophe Delage July 6th 2006
Background o Voronoi Diagram o Apollonius Diagram o Power Diagram o Incremental Algorithm o Invert and project
Voronoi Diagram
Apollonius Diagram Weighed point or Site s i is defined by p i, the point and w i the weight
Apollonius Graph Dual of Apollonius Diagram Vertex is a Site Edge two sites whose AW-Voronoi Cell share a face
Power Diagram
Incremental Algorithm The Three basic steps 1. Locate nearest neighbor 2. Check if new site is trivial 3. Update Vertex Conflict Edge Conflict
Vertex Conflict
Edge Conflict
Invert and Project
Invert and Project Convex Hull
Predicates Vertex Conflict, Edge Conflict Sub Predicates Orientation, Radical Side, Radical Intersection, Power Test, Order On a Line
Orientation
Radical Side
Radical Intersection
Vertex Conflict (revisited) There are 6 cases, and we have vertex conflict when: O-, RI-, RS{+|-} O+, RI+, RS+ O-, RI+, RS+
Vertex Conflict (degeneracies) When Orientation is 0 When Radical Intersection is 0
Pertubations Consistent ordering 1. Max Radius 2. Lexigraphically
Edge Conflict (revisited) We are looking for the case of q breaking 2 into multiple sections, when inverted and projection onto the unit sphere.
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
Further Work Optimize Edge Conflict Optimizations of the Incremental Algorithm Degeneracies in 3D Reduction of Exact Computations
Reducing the degree of Apollonius Diagram Predicates Merci!!!