Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Applied Geometry - IntroductionPlanar and Spatial Subdivisions

Gert Vegter

Applied Geometry, February 2010

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Survey

1 Planar and Spatial Subdivisions

2 Voronoi Diagrams

3 Delaunay Triangulations

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Animals

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Dry lakes

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Computable plant project

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Computable plant project

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Spatial Subdivision

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Sense of symmetry?

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Foam

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Foam architecture?

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Agriculture

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Geographic Information Systems

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Geographic Information Systems

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Geographic Information Systems

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Geographic modeling

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Multidisciplinary collaboration?

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Nearest neighbors of grid points

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Wave fronts

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi Diagrams

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi Diagrams

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi Diagrams

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi Diagrams

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi Diagrams

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Georgy Voronoi (1868–1908)

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi face

1

2

3

4

56

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi edge

1

2

3

4

56

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi vertex

1

2

3

4

56

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi Diagram

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi Diagram (2)

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi Diagram (3)

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi cells in 3D

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi vertices and circumscribed circles

1

2

3

4

5

6

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi vertices and circumscribed circles

1

2

3

4

5

6

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi vertices and circumscribed circles

1

2

3

4

5

6

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi vertices and circumscribed circles

1

2

3

4

5

6

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi vertices and circumscribed circles

1

2

3

4

5

6

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi vertices and circumscribed circles

1

2

3

4

5

6

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Delaunay triangulation

-1 1 2 3 4

1

2

3

4

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Boris Delaunay (1890–1980)

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Algorithm for Delaunay triangulation

Input: p[1], . . . , p[N] (points in the plane)

For each point-triple {p[i], p[j], p[k ]} do the followingCompute circumcircle CSet empty to trueFor all (other) points p[l] do

If p[l] inside C thenSet empty to false

If empty doAdd triangle p[i]p[j]p[k ] to triangulation

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Algorithm for Delaunay triangulation

Input: p[1], . . . , p[N] (points in the plane)

For each point-triple {p[i], p[j], p[k ]} do the followingCompute circumcircle CSet empty to trueFor all (other) points p[l] do

If p[l] inside C thenSet empty to false

If empty doAdd triangle p[i]p[j]p[k ] to triangulation

Number of computational steps?

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Algorithm for Delaunay triangulation

Input: p[1], . . . , p[N] (points in the plane)

For each point-triple {p[i], p[j], p[k ]} do the followingCompute circumcircle CSet empty to trueFor all (other) points p[l] do

If p[l] inside C thenSet empty to false

If empty doAdd triangle p[i]p[j]p[k ] to triangulation

Number of triples: ≈ 16 N3

Number of other points: N − 3

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Algorithm for Delaunay triangulation

Input: p[1], . . . , p[N] (points in the plane)

For each point-triple {p[i], p[j], p[k ]} do the followingCompute circumcircle CSet empty to trueFor all (other) points p[l] do

If p[l] inside C thenSet empty to false

If empty doAdd triangle p[i]p[j]p[k ] to triangulation

Number of triples: ≈ 16 N3

Number of other points: ≈ NHence: ≈ 1

6 N4 stepsApplied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Complexity of the Delaunay algorithm

Optimal Delaunay algorithm:approximately N log2 N steps.

N N log2 N 16N4

10 33 1667100 664 1.6 ∗ 107

1000 9966 1.6 ∗ 1011

10000 132877 1.6 ∗ 1015

100000 1.6 ∗ 106 1.6 ∗ 1019

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi Diagram and Delaunay triangulation

1

2

3

4

5

6

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi Diagram and Delaunay triangulation (2)

1

2

3

4

5

6

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Empty Delaunay triangles and circumcircles

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi and Delaunay cells in 3D

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi and Delaunay cells in 3D

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Voronoi and Delaunay cells in 3D

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Locally Delaunay Property

Locally Delaunay edge (left), non-locally Delaunay (right).

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Flip from ab to cd

a

b

c

d

cd locally Delaunay⇔ ab not locally Delaunay

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Incremental Delaunay algorithm

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Incremental Delaunay algorithm

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Incremental Delaunay algorithm

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Incremental Delaunay algorithm

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Incremental Delaunay algorithm

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Incremental Delaunay algorithm

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Maintain insertion history

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Maintain insertion history

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Maintain insertion history

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Maintain insertion history

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Connect the dots!

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Reconstruction from samples

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Reconstruction from samples (2)

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Reconstruction from samples

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Who is this?

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

David

Toen maakte zich David desmorgens vroeg op, en hij lietde schapen bij den hoeder,en hij nam het op, en ginghenen, gelijk als Isaï hembevolen had; en hij kwam aanden wagenburg, als het heirin slagorde uittoog, en menten strijde riep.

1 Samuel 17:20

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

The artist

Michelangelo Buonarotti(1475–1564)

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Scanning David

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

David’s head: sample

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

David’s head: sample

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

David’s head: reconstruction

Applied Geometry - Introduction (February 11, 2010)

Planar and Spatial SubdivisionsVoronoi Diagrams

Delaunay Triangulations

Art or Science?

Hyperbolic Geometry. . .

Applied Geometry - Introduction (February 11, 2010)