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

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.