Sequential Stochastic Construction of Random Polygons

摘要

Homogeneous Poisson fields of lines divide the plane into non-overlapping convex polygons. Of interest to researchers is geometrical probability have been the distributions of characteristics of the polygons induced by the distributions of the lines, especially N, the number of sides, S, the perimeter, and A, the area. A sequential stochastic process is developed from which an independent and identically distributed sample of polygons can be extracted with a stopping time. It is shown that the distribution of polygons so obtained is identical to the distribution of polygons in the Poisson field. The stochastic process is developed in full generality and can be applied to anisotropic cases as well as the case of most interest, the isotropic case. Useful families of anisotropic distributions for this problem are defined. The sequential stochastic process is used to derive general analytical expressions for polygon distributions for the investigation of the unknown distributions of N, S and A. Methods are also developed which provide the basis for very fast computer simulation of the process. A Monte Carlo study of distributions of N, S, and A in various cases is presented. In particular, a sample of 2,500,000 polygons in the isotropic case provides the most precise results to date. (Author)