]> Delaunay Triangulations
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Created 2005-06-24   Modified 2007-01-14
Chelton Evans

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Properties
2D in C++ O ( n 4 )
3D in C++ O ( n 5 )
Discussion of the Voronoi Diagram
Simplest DT Algorithm
<TODO>

Properties

V refers to the Voronoi diagram, D the Delaunay triangulation.

  1. V=dual(D), D=dual(V)
  2. face(D)--point(V)
  3. edge(D)--edge(V)
  4. point(D)--region(V)
  5. boundary(D) is the convex hull of the points in D
  6. interior of face(D) has no points in D
  7. region(V) is convex
  8. region(V) is unbounded if on convex hull
  9. point(V)--circle(D) where circle(D) has no points inside it from D
  10. pi is nearest to pj then edge(pi,pj) is an edge of D
  11. if circle through pi,pj contains no other points then edge(pi,pj) is an edge of D

Discussion of the Voronoi Diagram

Here is a Voronoi diagram with my strict Voronoi polygons.

diagg031.png

I have spent several hours looking at the Voronoi diagrams. The mathematics seems a little unfriendly so I did the usual thing and look at the properties to get a better feel and still I was not satisfied. So I implemented the algorithm and found a few interesting things.

Firstly consider only tessellation points inside the mesh and look at their associated Voronoi polygon. Iterating around the vertex produces this polygon and their is a one to one association with the triangle and the Voronoi point.

This was not mentioned in J.O'Rouke's detailed properties of DT in Computational Geometry in C. Indeed what got me interested was looking at Voronoi diagrams as they appeared slightly incorrect.

Lets define a strict Voronoi polygon to have the same number of points as the number of triangles rotated about the associated DT point. Then looking at Voronoi diagrams you realize that the polygons with vertexes on the boundary have one more edge than their number of triangles.

If we use my strict Voronoi polygon the outer regions overlap. What people do is consider where the two lines meet and create a new line from there extending out and in the middle of the other two. This lets the space be uniquely partitioned into Voronoi regions.

Simplest DT Algorithm

It was suggested to me that I should implement the simplest DT where you go through all the combination of points that make a triangle and verify that no other point is inside the circle through the triangle.

This is generalized in 3D to find the sphere through four points and if no other point in the tessellation is inside the sphere then the four points make a tetrahedron that is in the DT tessellation.

See Sphere Through Four Points on the maths.

<TODO>