Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary Partitions of Line Segments in the
Plane
Mathieu Brévilliers, Laboratoire MIA, UHA
Point set triangulations
Mathieu Brévilliers, Laboratoire MIA, UHA
Point set triangulations
Mathieu Brévilliers, Laboratoire MIA, UHA
Delaunay triangulations
Mathieu Brévilliers, Laboratoire MIA, UHA
Duality
Mathieu Brévilliers, Laboratoire MIA, UHA
Legal edge property
Mathieu Brévilliers, Laboratoire MIA, UHA
Legal edge property
The Delaunay triangulation is the unique triangulation without illegal edge.
O(n) checker for Delaunay triangulations
Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary partitions
Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary partitions
Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary partitions
Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary partitions
Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary partitions
Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary partitions
Mathieu Brévilliers, Laboratoire MIA, UHA
Shape of edges
Mathieu Brévilliers, Laboratoire MIA, UHA
Shape of edges
Mathieu Brévilliers, Laboratoire MIA, UHA
Shape of edges
Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
Mathieu Brévilliers, Laboratoire MIA, UHA
Topology
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
Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary Delaunay partition
Mathieu Brévilliers, Laboratoire MIA, UHA
Elementary Delaunay partition
Mathieu Brévilliers, Laboratoire MIA, UHA
Duality
Mathieu Brévilliers, Laboratoire MIA, UHA
Legal edge property
Mathieu Brévilliers, Laboratoire MIA, UHA
Legal edge property
Legal Illegal
Mathieu Brévilliers, Laboratoire MIA, UHA
Legal edge property
The elementary Delaunay partition is the unique elementary partition without illegal edge.
Mathieu Brévilliers, Laboratoire MIA, UHA
Checker for Delaunay topologyTopology Geometry with faces in
Delaunay positions
Is it an elementary partition ?
Mathieu Brévilliers, Laboratoire MIA, UHA
Checker for Delaunay topology
a
b c
d
a
bc
d
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
Mathieu Brévilliers, Laboratoire MIA, UHA
Future works
Flip algorithm
Equiangularity for elementary Delaunay partitions
More general set of sites
Higher dimensions
Mathieu Brévilliers, Laboratoire MIA, UHA
Thank you!