elementary partitions of line segments in the plane

Mathieu Brévilliers, Laboratoire MIA, UHA Elementary Partitions of Line Segments in the Plane

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Elementary Partitions of Line Segments in the Plane. Point set triangulations. Point set triangulations. Delaunay triangulations. Duality. Legal edge property. Legal edge property. The Delaunay triangulation is the unique triangulation without illegal edge. - PowerPoint PPT Presentation

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Mathieu Brévilliers, Laboratoire MIA, UHA

Elementary Partitions of Line Segments in the

Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Point set triangulations

Page 3: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Point set triangulations

Page 4: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Delaunay triangulations

Page 5: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Duality

Page 6: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Legal edge property

Page 7: Elementary Partitions of Line Segments in the Plane

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

Page 15: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Shape of edges

Page 16: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Shape of edges

Page 17: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Topology

Page 18: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Topology

Page 19: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Topology

Page 20: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Topology

Page 21: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Topology

Page 22: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Topology

Page 23: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Topology

Page 24: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Topology

Page 25: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Topology

Page 26: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Topology

Page 27: Elementary Partitions of Line Segments in the Plane

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

Page 28: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Elementary Delaunay partition

Page 29: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Elementary Delaunay partition

Page 30: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Duality

Page 31: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Legal edge property

Page 32: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Legal edge property

Legal Illegal

Page 33: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Legal edge property

The elementary Delaunay partition is the unique elementary partition without illegal edge.

Page 34: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Checker for Delaunay topologyTopology Geometry with faces in

Delaunay positions

Is it an elementary partition ?

Page 35: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Checker for Delaunay topology

b c

Page 36: Elementary Partitions of Line Segments in the Plane

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

Page 37: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Future works

Flip algorithm

Equiangularity for elementary Delaunay partitions

More general set of sites

Higher dimensions

Page 38: Elementary Partitions of Line Segments in the Plane

Mathieu Brévilliers, Laboratoire MIA, UHA

Thank you!

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elementary partitions of line segments in the plane

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