The proportion of triangles in a class of anisotropic Poisson line tessellations

TL;DR

This paper investigates the proportion of triangular cells in anisotropic Poisson line tessellations with multiple directions, providing new insights and deviations from classical isotropic results.

Contribution

It introduces a method to determine the probability of triangular cells in anisotropic Poisson line tessellations with multiple directions, extending classical isotropic results.

Findings

Derived the proportion of triangles in anisotropic tessellations.

Obtained a new deviation of Miles' classical result.

Provided approximation methods for non-isotropic cases.

Abstract

Stationary Poisson processes of lines in the plane are studied whose directional distributions are concentrated on $k \ge 3$ equally spread directions. The random lines of such processes decompose the plane into a collection of random polygons, which form a so-called Poisson line tessellation. The focus of this paper is to determine the proportion of triangles in such tessellations, or equivalently, the probability that the typical cell is a triangle. As a by-product, a new deviation of Miles' classical result for the isotropic case is obtained by an approximation argument.