Voronoi diagram

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}}

.