Finding the Area and Perimeter Distributions for Flat Poisson Processes of a Straight Line and Voronoi Diagrams

Kanel-Belov, A.Y.; Golafshan, M.; Malev, S.G.; Yavich, R.P.

Авторы: Kanel-Belov A.Ya, Golafshan M., Malev S.G., Yavich R.P.
Журнал: Doklady Mathematics
Год издания: 2024
Издательство: Maik Nauka/Interperiodica Publishing
Местоположение издательства: Russian Federation
Аннотация: 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 classicalproblem in statistical geometry. Moments for such distributions have been investigated since 1972 [1]. We givea complete solution of these problems for the plane, as well as for Voronoi diagrams. The following problemsare solved: 1. A random set of straight lines is given on the plane, all shifts are equiprobable, and the distribution law has the form What is the area (perimeter) distribution of the parts of the partition? 2. A randomset of points is marked on the plane. Each point A is associated with a “region of attraction,” which is a set ofpoints on the plane to which A is the closest of the marked set. The idea is to interpret a random polygon asthe evolution of a segment on a moving one and construct kinetic equations. It is sufficient to take intoaccount a limited number of parameters: the covered area (perimeter), the length of the segment, and theangles at its ends. We show how to reduce these equations to the Riccati equation using the Laplace transform.Keywords: geometric probabilities, Poisson line process, Voronoi diagram, kinetic equation, Markov equation, random sets, statistical geometry, distributions of random variables
Добавил в систему: Белов Алексей Яковлевич

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Finding the Area and Perimeter Distributions for Flat Poisson Processes of a Straight Line and Voronoi Diagramsстатья

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