Delaunay simplexes

An example of Delaunay triangulation in two-dimensions

A Delaunay simplex is the dual of the Voronoi diagram. Delaunay simplexes were developed by Борис Николаевич Делоне. In two-dimensions $({\mathbb {R} }^{2})$ it is more commonly known as Delaunay triangulation, and in three-dimensions $({\mathbb {R} }^{3})$, as Delaunay tetrahedralisation.

A Delaunay triangulation fulfils the empty circle property (also called Delaunay property): the circumscribing circle of any facet of the triangulation contains no data point in its interior. For a point set with no subset of four co-circular points the Delaunay triangulation is unique. A similar property holds for tetrahedralisation in three dimensions.

References

1. Математические основы структурного анализа кристаллов (совместно с А.Д.Александровым и Н.Падуровым), Москва, Матем. литература, 1934 г.
2. A. V. Anikeenko, M. L. Gavrilova and N. N. Medvedev "A Novel Delaunay Simplex Technique for Detection of Crystalline Nuclei in Dense Packings of Spheres", Lecture Notes in Computer Science 3480 pp. 816-826 (2005)

External links

• The CGAL project on computational geometry