reducing the degree of apollonius diagram predicates david millman advisors: sylvain pion and...
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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
Given Points pi and m,
Euclidian distance:d p
i,m( ) = pi -m
Given P = p1…pn{ }The Voronoi cell of pi:
V pi( ) = m∈° d ∀j,d pi,m( ) ≤d pj,m( ){ }
Apollonius Diagram
Weighed point or Site si is defined by pi, the point and wi the weight
Given Site si and Point m,
weighted distance:d+ si,m( ) = pi -m -wi
Given S = s1…sn{ }Additively weighted Voronoi cell of si:V si( ) = m∈° d ∀j,d+ si,m( ) ≤d+ s j,m( ){ }
Apollonius Graph
• Dual of Apollonius Diagram
• Vertex is a Site
• Edge two sites whose AW-Voronoi Cell share a face
Power Diagram
Given stie σ i and x, Power of x:
dP σ i,x( ) = x−ci( )2−ri
2
Given Σ = σ1…σn{ }The Power cell σ i:
Ρ σ i( ) = x∈° d ∀j,dP ci,x( ) ≤dP c j,x( ){ }
Weighed point or Site σ i is defined by ci, the center and ri the radius
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
Invert and Project Convex Hull
Predicates
• PredicatesVertex Conflict, Edge Conflict
• Sub PredicatesOrientation, 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
Grow the largest site by +ε
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!!!