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Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary Partitions of Line Segments in the
Plane
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Mathieu Brévilliers, Laboratoire MIA, UHA
Point set triangulations
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Mathieu Brévilliers, Laboratoire MIA, UHA
Point set triangulations
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Mathieu Brévilliers, Laboratoire MIA, UHA
Delaunay triangulations
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Mathieu Brévilliers, Laboratoire MIA, UHA
Duality
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Mathieu Brévilliers, Laboratoire MIA, UHA
Legal edge property
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Mathieu Brévilliers, Laboratoire MIA, UHA
Legal edge property
The Delaunay triangulation is the unique triangulation without illegal edge.
O(n) checker for Delaunay triangulations
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Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary partitions
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Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary partitions
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Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary partitions
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Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary partitions
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Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary partitions
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Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary partitions
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Mathieu Brévilliers, Laboratoire MIA, UHA
Shape of edges
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Mathieu Brévilliers, Laboratoire MIA, UHA
Shape of edges
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Mathieu Brévilliers, Laboratoire MIA, UHA
Shape of edges
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Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
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Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
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Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
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Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
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Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
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Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
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Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
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Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
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Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
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Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
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Mathieu Brévilliers, Laboratoire MIA, UHA
Numbers of edges and faces
3n – n’ – 3 edges
2n – n’ – 2 faces
n : number of sites n’ : number of sides of the CH that are not sites
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Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary Delaunay partition
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Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary Delaunay partition
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Mathieu Brévilliers, Laboratoire MIA, UHA
Duality
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Mathieu Brévilliers, Laboratoire MIA, UHA
Legal edge property
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Mathieu Brévilliers, Laboratoire MIA, UHA
Legal edge property
Legal Illegal
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Mathieu Brévilliers, Laboratoire MIA, UHA
Legal edge property
The elementary Delaunay partition is the unique elementary partition without illegal edge.
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Mathieu Brévilliers, Laboratoire MIA, UHA
Checker for Delaunay topologyTopology Geometry with faces in
Delaunay positions
Is it an elementary partition ?
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Mathieu Brévilliers, Laboratoire MIA, UHA
Checker for Delaunay topology
a
b c
d
a
bc
d
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Mathieu Brévilliers, Laboratoire MIA, UHA
Checker for Delaunay topology
1. For each edge, the test runs in constant time
2. O(n) edges in an elementary partition
Linear algorithm
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Mathieu Brévilliers, Laboratoire MIA, UHA
Future works
Flip algorithm
Equiangularity for elementary Delaunay partitions
More general set of sites
Higher dimensions
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Mathieu Brévilliers, Laboratoire MIA, UHA
Thank you!