# Voronoi diagram

A very important geometric structure in computational geometry, named after G.F. Voronoi. The earliest significant use of Voronoi diagrams seems to have occurred in the work of C.F. Gauss, P.G.L. Dirichlet and Voronoi on the reducibility of positive-definite quadratic forms (cf. Quadratic form).

Let $ S = \{ p _ {1} \dots p _ {n} \} $ be a set of $ n $ points in $ \mathbf R ^ {d} $. The Voronoi diagram generated by $ S $ is the partition of the $ \mathbf R ^ {d} $ into $ n $ convex cells, the Voronoi cells, $ V _ {i} $, where each $ V _ {i} $ contains all points of $ \mathbf R ^ {d} $ closer to $ p _ {i} $ than to any other point:

$$ V _ {i} = \left \{ x : {\forall j \neq i, d ( x,p _ {i} ) \leq d ( x,p _ {j} ) } \right \} , $$

where $ d ( x,y ) $ is the Euclidean distance between $ x $ and $ y $.

See also Delaunay triangulation.

#### References

[a1] | F.P. Preparata, M.I. Shamos, "Computational geometry: an introduction" , Springer (1985) MR805539 Zbl 0575.68059 Zbl 0759.68037 |

[a2] | H. Edelsbrunner, "Algorithms in combinatorial geometry" , Springer (1987) MR0904271 Zbl 0634.52001 |

[a3] | A. Okabe, B. Boots, K. Sugihara, "Spatial tessellations: concepts and applications of Voronoi diagrams" , Wiley (1992) MR1210959 Zbl 0877.52010 |

[a4] | G.F. Voronoi, "Nouvelles applications des parametres continus a la theorie des formes quadratiques" J. Reine Angew. Math. , 134 (1908) pp. 198–287 Zbl 38.0261.01 Zbl 39.0274.01 |

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Voronoi diagram.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Voronoi_diagram&oldid=49164