Random Processes of Linear Segments and Graphs

Summary

By a graph G we understand a finite set of points (vertices) together with the line segments which united some pains of distinct points of the set. Sets of congruent graphs are considered. The position of a graph on the plane is defined by the position of one of its vertices P and a notation ⌽ about P. Assuming P Poisson distributed on the plane and ⌽ uniformly distributed oven 0 ≤ ⌽ < 2_π, we extend to graph processes some known propenties of Line segment processes (Coleman [1], [2]; Parker and Cowan [3] ). We find the probability that the distance from a point chosen at random independently of the process of graphs to the nearest vertice of a graph on to the nearest graph exceeds u. Some of the results are also extended from the euclidean plane to surfaces (sets of geodesic segments and sets of geodesic graphs), for instance to the sphere and to the hyperbolic plane.