Vertex number of the typical cell in a tri-directional Poisson line tessellation

TL;DR

This paper analyzes the typical cell in a tri-directional Poisson line tessellation, explicitly determining the probabilities of the cell being a triangle, quadrilateral, pentagon, or hexagon based on directional weights.

Contribution

It provides explicit probability formulas for the shape of the typical cell in a tri-directional Poisson line tessellation, including extremal cases.

Findings

Probabilities for each polygon shape are explicitly derived.

The shape probabilities depend on the weights of the directional distribution.

Extremal cases of the shape probabilities are discussed.

Abstract

This paper deals with the typical cell in a Poisson line tessellation in the plane whose directional distribution is concentrated on three equally spread values with possibly different weights. Such a random polygon can only be a triangle, a quadrilateral, a pentagon or a hexagon. The probability for each of these cases is determined explicitly in terms of the weights. Extremal cases are discussed as well.