Finding the area and perimeter distributions for flat Рoisson processes of a straight line and Voronoi diagrams

The study of distribution functions (with respect to areas, perimeters) for partitioning a plane (space) by a random field of straight lines (hyperplanes) and for obtaining Voronoi diagrams is a classical problem in statistical geometry. Moments for such distributions have been investigated since 1972 [1]. We give a complete solution of these problems for the plane, as well as for Voronoi diagrams. The following problems are solved:
1. A random set of straight lines is given on the plane, all shifts are equiprobable, and the distribution law has the form $F(\varphi)$. What is the area (perimeter) distribution of the parts of the partition?
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 A is the closest of the marked set.
The idea is to interpret a random polygon as the evolution of a segment on a moving one and construct kinetic equations. It is sufficient to take into account a limited number of parameters: the covered area (perimeter), the length of the segment, and the angles at its ends. We show how to reduce these equations to the Riccati equation using the Laplace transform.