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Voronoi diagram
In mathematics, a Voronoi diagram is a partitioning of a plane into regions based on distance to points in a specific subset of the plane. That set of points (called seeds, sites, or generators) is specified beforehand, and for each seed there is a corresponding region consisting of all points closer to that seed than to any other. These regions are called Voronoi cells. The Voronoi diagram of a set of points is dual to its Delaunay triangulation. Put simply, it's a diagram created by taking pairs of points that are close together and drawing a line that is equidistant between them and perpendicular to the line connecting them. That is, all points on the lines in the diagram are equidistant to the nearest two (or more) source points.
It is named after Georgy Voronoy, and is also called a Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi diagrams have practical and theoretical applications to a large number of fields, mainly in science and technology but also including visual art.〔Franz Aurenhammer (1991). ''Voronoi Diagrams – A Survey of a Fundamental Geometric Data Structure''. ACM Computing Surveys, 23(3):345–405, 1991〕〔Atsuyuki Okabe, Barry Boots, Kokichi Sugihara & Sung Nok Chiu (2000). ''Spatial Tessellations – Concepts and Applications of Voronoi Diagrams''. 2nd edition. John Wiley, 2000, 671 pages ISBN 0-471-98635-6〕
==The simplest case==
In the simplest and most familiar case (shown in the first picture), we are given a finite set of points in the Euclidean plane. In this case each site ''p''''k'' is simply a point, and its corresponding Voronoi cell (also called Voronoi region or Dirichlet cell) ''R''''k'' consists of every point whose distance to ''p''''k'' is less than or equal to its distance to any other ''p''''k''. Each such cell is obtained from the intersection of half-spaces, and hence it is a convex polygon. The line segments of the Voronoi diagram are all the points in the plane that are equidistant to the two nearest sites. The Voronoi vertices (nodes) are the points equidistant to three (or more) sites.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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