Voronoi Diagrams

Synonyms

Dirichlet tessellation; Voronoi decomposition; Voronoi tessellation; Thiessen polygons

Definition

The Voronoi diagram of a given set $P=\left\{{p}_1,\dots, {p}_n\right\}$ of n points in ℝ^d partitions the space of ℝ^d into n regions [2]. Each region includes all points in ℝ^d with a common closest point in the given set P according to a distance metric D(.,.). That is, the region corresponding to the point p ∈ P contains all the points q ∈ ℝ^d for which the following holds:

$$\forall p' \in P'\mathrel{\not =}p,\ D(q,p) \leq D(q,p')$$

The equality holds for the points on the borders of p’s and p’s regions. Incorporating arbitrary distance metrics D(.,.) results in different variations of Voronoi diagrams. As an example, additively weighted Voronoi diagrams are defined by using the distance metric $D\left(p,q\right)={L}_2\left(p,q\right)+w(p)$ where L₂(.,.) is the Euclidean distance and w(p) is a numeric weight assigned to p. A thorough discussion on all variations is...