Reducing the degree of Apollonius Diagram Predicates

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Reducing the degree of Apollonius Diagram
Predicates

• David Millman
• Advisors Sylvain Pion and Christophe Delage
• July 6th 2006

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Background

• Voronoi Diagram
• Apollonius Diagram
• Power Diagram
• Incremental Algorithm
• Invert and project

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Voronoi Diagram
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Apollonius Diagram

• Weighed point or Site si is defined by pi, the
  point and wi the weight

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Apollonius Graph

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

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Power Diagram
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Incremental Algorithm

• The Three basic steps
• Locate nearest neighbor
• Check if new site is trivial
• Update
• Vertex Conflict
• Edge Conflict

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Vertex Conflict
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Edge Conflict
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Invert and Project
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Invert and Project
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Invert and Project Convex Hull
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Predicates

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

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Orientation
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Radical Side
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Radical Intersection
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Vertex Conflict (revisited)

• There are 6 cases, and we have vertex conflict
  when

O-, RI-, RS-
O, RI, RS
O-, RI, RS
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Vertex Conflict (degeneracies)

• When Orientation is 0
• When Radical Intersection is 0

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Pertubations

• Consistent ordering
• Max Radius
• Lexigraphically

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Edge Conflict (revisited)

• We are looking for the case of q breaking 2 into
  multiple sections, when inverted and projection
  onto the unit sphere.

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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

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Further Work

• Optimize Edge Conflict
• Optimizations of the Incremental Algorithm
• Degeneracies in 3D
• Reduction of Exact Computations

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Reducing the degree of Apollonius Diagram
Predicates
Merci!!!