Contact and chord length distributions of the Poisson Voronoi tessellation

Abstract

This paper presents the form of some characteristics of the Voronoi tessellation which is generated by a stationary Poisson process in . Expressions are given for the spherical and linear contact distribution functions. These formulae lead to numerically tractable double-integral formulae for chord length probability density functions.

Random Laguerre tessellations

“…This generalizes results by Muche and Stoyan (1992). Tessellations with respect to more general point processes of generators, in the Voronoi case studied by Heinrich (1998), were not considered in this context.…”

“…Later also formulas for several distribution functions became available. For example, Muche and Schlather studied the edge length distribution function (Muche, 1996b(Muche, , 2005Schlather, 2000), Muche and Stoyan (1992) considered contact and chord length distributions, and Calka (2003b) studied the distribution of the number of faces and of the area of a planar Voronoi tessellation. A summary of results can be found in the book by Okabe et al (2000).…”

“…Contact and chord length distributions of the Poisson Voronoi tessellation have been studied by Muche and Stoyan (1992) while Heinrich (1998) investigated the Voronoi tessellation with respect to more general point processes. Here, we are going to consider Poisson Laguerre tessellations.…”

On the Dilated Facets of a Poisson-Voronoi Tessellation

“…Then the contact distribution function H B of Θ is defined via Explicit formulae for contact and chord length distribution functions of the Poisson-Voronoi tessellation are given in Heinrich (1998) and Muche and Stoyan (1992). Theorem 1.…”

“…For d = 2 and d = 3 the spherical contact distribution function of Ξ has been computed explicitly in Muche and Stoyan (1992)…”