Delaunay Triangulation and Randomized Constructions

Years and Authors of Summarized Original Work

• 1989; Clarkson, Shor

• 1993; Seidel

• 2002; Devillers

• 2003;Amenta, Choi, Rote

Problem Definition

The Delaunay triangulation and the Voronoi diagram are two classic geometric structures in the field of computational geometry. Their success can perhaps be attributed to two main reasons: Firstly, there exist practical, efficient algorithms to construct them; and secondly, they have an enormous number of useful applications ranging from meshing and 3D-reconstruction to interpolation.

Given a set S of n sites in some space $\mathbb{E}$, we define the Voronoi regionV _S (p) of p ∈ S to be the set of points in $\mathbb{E}$ whose nearest neighbor in S is p (for some distance δ):

$$\displaystyle\begin{array}{rcl} V (p) = \left \{x\,\in \,\mathbb{E},\forall q\,\in \,S\setminus \{p\}\delta (x,p) <\delta (x,q)\right \}.& & {}\\ \end{array}$$

It is easily seen that these regions form a partition of $\mathbb{E}$ into convex regions which we refer to as cells...