Statistics of cross sections of Voronoi tessellations

TL;DR

This paper analyzes the statistical properties of cross sections of 3D Voronoi tessellations, deriving analytical formulas for radii and areas of sections, and validating them with numerical simulations.

Contribution

It introduces analytical expressions for the distributions of section radii and areas in Voronoi tessellations, approximating cells as spheres for tractability.

Findings

Derived the probability density function for section radii involving Meijer G-function.

Computed the probability density function for cross-sectional areas.

Validated analytical results with numerical simulations.

Abstract

In this paper we investigate relationships between the volumes of cells of three-dimensional Voronoi tessellations and the lengths and areas of sections obtained by intersecting the tessellation with a randomly oriented plane. Here, in order to obtain analytical results, Voronoi cells are approximated to spheres. First, the probability density function for the lengths of the radii of the sections is derived and it is shown that it is related to the Meijer $G$-function; its properties are discussed and comparisons are made with the numerical results. Next the probability density function for the areas of cross sections is computed and compared with the results of numerical simulations.