In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed there is a corresponding region, called a Voronoi cell comprising all points of the plane closer to that seed than to any other. The Voronoi diagram of a set of points is dual to that set's Delaunay triangulation. The Voronoi diagram is named after mathematician 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 cells are also known as Thiessen polygons, after Alfred H. Thiessen. Voronoi diagrams have practical and theoretical applications in many fields, mainly in science and technology, but also in visual art.
Simplest case In the simplest case, shown in the first picture, we are given a finite set of points { p 1 , … p n } {\displaystyle \{p_{1},\dots p_{n}\}} in the Euclidean plane. In this case, each point p k {\displaystyle p_{k}} has a corresponding cell R k {\displaystyle R_{k}} consisting of the points in the Euclidean plane for which p k {\displaystyle p_{k}} is the nearest site: the distance to p k {\displaystyle p_{k}} is less than or equal to the minimum distance to any other site p j {\displaystyle p_{j}} . For one other site p j {\displaystyle p_{j}} , the points that are closer to p k {\displaystyle p_{k}} than to p j {\displaystyle p_{j}} , or equally distant, form a closed half-space, whose boundary is the perpendicular bisector of line segment p j p k {\displaystyle p_{j}p_{k}} . Cell R k {\displaystyle R_{k}} is the intersection of all of these n − 1 {\displaystyle n-1} half-spaces, and hence it is a convex polygon. When two cells in the Voronoi diagram share a boundary, it is a line segment, ray, or line, consisting of all the points in the plane that are equidistant to their two nearest sites. The vertices of the diagram, where three or more of these boundaries meet, are the points that have three or more equally distant nearest sites.
Formal definition Let X {\textstyle X} be a metric space with distance function d {\textstyle d} . Let K {\textstyle K} be a set of indices and let ( P k ) k ∈ K {\textstyle (P_{k})_{k\in K}} be a tuple (indexed collection) of nonempty subsets (the sites) in the space X {\textstyle X} . The Voronoi cell, or Voronoi region, R k {\textstyle R_{k}} , associated with the site P k {\textstyle P_{k}} is the set of all points in X {\textstyle X} whose distance to P k {\textstyle P_{k}} is not greater than their distance to the other sites P j {\textstyle P_{j}} , where j {\textstyle j} is any index different from k {\textstyle k} . In other words, if d ( x , A ) = inf { d ( x , a ) ∣ a ∈ A } {\textstyle d(x,\,A)=\inf\{d(x,\,a)\mid a\in A\}} denotes the distance between the point x {\textstyle x} and the subset A {\textstyle A} , then
R k = { x ∈ X ∣ d ( x , P k ) ≤ d ( x , P j ) for all j ≠ k } {\displaystyle R_{k}=\{x\in X\mid d(x,P_{k})\leq d(x,P_{j})\;{\text{for all}}\;j\neq k\}}
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