Finding the area and perimeter distributions for flat Poisson processes of a straight line and Voronoi mosaics

The study of distribution functions (by areas, perimeters) for partitioning a plane (space) by a random field of straight lines (hyperplanes) and for Voronoi mosaics is a classical problem of statistical geometry. Starting from 1972 [1] to the present, moments for such distributions have been investigated. We give a complete solution of these problems for the plane, as well as for Voronoi mosaics. We investigate the following tasks:

1. A random set of straight lines is given on the plane, all shifts are equally probable, and the distribution law has the form F(φ). What is the distribution of the parts of the partition by areas (perimeters)?
2. A random set of points is marked on the plane. Each point A is associated with a “region of attraction”, which is a set of points on the plane to which the point A is the closest of the set marked.

The idea is to interpret a random polygon as the evolution of a segment on a moving one and construct kinetic equations. At the same time, it is sufficient to take into account a limited number of parameters: the area covered (perimeter), the length of the segment, the angles at its ends. We will show how to reduce these equations to the Riccati equation using the Laplace transform. (see theorems 1, 1 and 2).