# Difference between revisions of "Voronoi cells"

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The '''Voronoi cells''' or '''Voronoi tessellation''' or '''Dirichlet tesselation''' or '''Wigner-Seitz cells'''. | The '''Voronoi cells''' or '''Voronoi tessellation''' or '''Dirichlet tesselation''' or '''Wigner-Seitz cells'''. | ||

− | + | This is the diagram that results when a cell is defined around each of the points (or nodes, | |

− | + | or vertices) of a network with the following criterion: each point in the cell is closer to its node | |

+ | than to any of the others. This very intuitive partition of space results in the Voronoi tessellation. The | ||

+ | typical example is to, e.g., assign areas of a country to different fire stations, so that if a fire | ||

+ | occurs, the corresponding station is the closest one. | ||

+ | |||

+ | Voronoi cells are '''dual''' of [[Delaunay simplexes]]. | ||

+ | |||

+ | |||

==External links== | ==External links== | ||

*[http://www.cgal.org/Manual/3.3/doc_html/cgal_manual/packages.html#part_IX The CGAL project on computational geometry] | *[http://www.cgal.org/Manual/3.3/doc_html/cgal_manual/packages.html#part_IX The CGAL project on computational geometry] |

## Revision as of 12:26, 11 October 2007

The **Voronoi cells** or **Voronoi tessellation** or **Dirichlet tesselation** or **Wigner-Seitz cells**.
This is the diagram that results when a cell is defined around each of the points (or nodes,
or vertices) of a network with the following criterion: each point in the cell is closer to its node
than to any of the others. This very intuitive partition of space results in the Voronoi tessellation. The
typical example is to, e.g., assign areas of a country to different fire stations, so that if a fire
occurs, the corresponding station is the closest one.

Voronoi cells are **dual** of Delaunay simplexes.

## External links

## References

- G. F. Voronoi "Nouvelles applications des paramètres continus à la théorie des formes quadratiques - Deuxième mémoire", Journal für die reine und angewandte Mathematik
**134**pp. 198-287 (1908) - Witold Brostow, Jean-Pierre Dussault and Bennett L. Fox "Construction of Voronoi polyhedra", Journal of Computational Physics
**29**pp. 81-92 (1978) - J. L. Finney "A procedure for the construction of Voronoi polyhedra", Journal of Computational Physics
**32**pp. 137-143 (1979) - Masaharu Tanemura, Tohru Ogawa and Naofumi Ogita "A new algorithm for three-dimensional voronoi tessellation", Journal of Computational Physics
**51**pp. 191-207 (1983) - C. S. Hsu and Aneesur Rahman "Interaction potentials and their effect on crystal nucleation and symmetry", Journal of Chemical Physics
**71**pp. 4974-4986 (1979) - J. Neil Cape, John L. Finney and Leslie V. Woodcock "An analysis of crystallization by homogeneous nucleation in a 4000-atom soft-sphere model", Journal of Chemical Physics
**75**pp. 2366-2373 (1981) - Nikolai N. Medvedev, Alfons Geiger and Witold Brostow "Distinguishing liquids from amorphous solids: Percolation analysis on the Voronoi network", Journal of Chemical Physics
**93**pp. 8337-8342 (1990) - J. C. Gil Montoro and J. L. F. Abascal "The Voronoi polyhedra as tools for structure determination in simple disordered systems", Journal of Physical Chemistry
**97**pp. 4211 - 4215 (1993) - V. Senthil Kumar and V. Kumaran "Voronoi cell volume distribution and configurational entropy of hard-spheres", Journal of Chemical Physics
**123**114501 (2005) - V. Senthil Kumar and V. Kumaran "Voronoi neighbor statistics of hard-disks and hard-spheres", Journal of Chemical Physics
**123**074502 (2005)