Voronoi cells: Difference between revisions

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A '''Voronoi cell''' <ref>[http://dx.doi.org/10.1515/crll.1908.134.198 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)]</ref>(also known as: '''Voronoi polyhedra''',  '''Voronoi tessellations''', '''Dirichlet tesselations''', '''Wigner-Seitz cells''') 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 lattice | dual]] of [[Delaunay simplexes]].
==Algorithms==
*[http://dx.doi.org/10.1016/0021-9991(78)90110-9 Witold Brostow, Jean-Pierre Dussault and Bennett L. Fox "Construction of Voronoi polyhedra", Journal of Computational Physics  '''29''' pp. 81-92 (1978)]
*[http://dx.doi.org/10.1016/0021-9991(79)90146-3  J. L. Finney "A procedure for the construction of Voronoi polyhedra", Journal of Computational Physics  '''32''' pp. 137-143 (1979)]
*[http://dx.doi.org/10.1016/0021-9991(83)90087-6  Masaharu Tanemura, Tohru Ogawa and Naofumi Ogita "A new algorithm for three-dimensional voronoi tessellation", Journal of Computational Physics  '''51''' pp. 191-207 (1983)]
*[http://dx.doi.org/10.1016/0021-9991(86)90123-3  N. N. Medvedev  "The algorithm for three-dimensional voronoi polyhedra", Journal of Computational Physics '''67''' pp. 223-229 (1986)]
==References==
==References==
# G.F. Voronoi. Z. Reine Angew. Math. '''134'''  pp. 198- (1908)
<references/>
#[http://dx.doi.org/10.1016/0021-9991(78)90110-9 Witold Brostow, Jean-Pierre Dussault and Bennett L. Fox "Construction of Voronoi polyhedra", Journal of Computational Physics  '''29''' pp. 81-92 (1978)]
'''Related reading'''
#[http://dx.doi.org/10.1016/0021-9991(79)90146-3  J. L. Finney "A procedure for the construction of Voronoi polyhedra", Journal of Computational Physics  '''32''' pp. 137-143 (1979)]
*[http://dx.doi.org/10.1063/1.438311    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)]
#[http://dx.doi.org/10.1016/0021-9991(83)90087-6  Masaharu Tanemura, Tohru Ogawa and Naofumi Ogita "A new algorithm for three-dimensional voronoi tessellation", Journal of Computational Physics  '''51''' pp. 191-207 (1983)]
*[http://dx.doi.org/10.1063/1.442299 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)]
#[http://dx.doi.org/10.1063/1.438311    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)]
*[http://dx.doi.org/10.1063/1.459711 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)]
#[http://dx.doi.org/10.1063/1.442299 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)]
*[http://dx.doi.org/10.1021/j100118a044 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)]
#[http://dx.doi.org/10.1063/1.459711 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)]
*[http://dx.doi.org/10.1063/1.2011390      V. Senthil Kumar and V. Kumaran "Voronoi cell volume distribution and configurational entropy of hard-spheres", Journal of Chemical Physics '''123''' 114501 (2005)]
#[http://dx.doi.org/10.1021/j100118a044 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)]
*[http://dx.doi.org/10.1063/1.2000233    V. Senthil Kumar and V. Kumaran  "Voronoi neighbor statistics of hard-disks and hard-spheres", Journal of Chemical Physics '''123''' 074502 (2005)]
#[http://dx.doi.org/10.1063/1.2011390      V. Senthil Kumar and V. Kumaran "Voronoi cell volume distribution and configurational entropy of hard-spheres", Journal of Chemical Physics '''123''' 114501 (2005)]
*[http://dx.doi.org/10.1063/1.3382485  Jagtar Singh Hunjan and Byung Chan Eu "The Voronoi volume and molecular representation of molar volume: Equilibrium simple fluids", Journal of Chemical Physics '''132''' 134510 (2010)]
#[http://dx.doi.org/10.1063/1.2000233    V. Senthil Kumar and V. Kumaran  "Voronoi neighbor statistics of hard-disks and hard-spheres", Journal of Chemical Physics '''123''' 074502 (2005)]
==External links and resources==
[[category: Computer simulation techniques]]
*[http://www.cgal.org/Manual/3.3/doc_html/cgal_manual/packages.html#part_IX The CGAL project on computational geometry]
*[ftp://ftp.dl.ac.uk/ccp5/ALLEN_TILDESLEY/F.35  The Voronoi construction in 2d and 3d] sample FORTRAN computer code from the book [http://www.oup.com/uk/catalogue/?ci=9780198556459 M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", Oxford University Press (1989)].
[[category: Computer simulation techniques ]] [[category: Mathematics ]]

Latest revision as of 10:32, 7 April 2010

A Voronoi cell [1](also known as: Voronoi polyhedra, Voronoi tessellations, Dirichlet tesselations, Wigner-Seitz cells) 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.

Algorithms[edit]

References[edit]

  1. ↑ 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)

Related reading

External links and resources[edit]

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