Averages for polygons formed by random lines in euclidean and hyperbolic planes

Abstract

We consider a countable number of independent random uniform lines in the hyperbolic plane (in the sense of the theory of geometrical probability) which divide the plane into an infinite number of convex polygonal regions. The main purpose of the paper is to compute the mean number of sides, the mean perímeter, the mean area and the second order moments of these quantities of such polygonal regions. For the Euclidean plane the problema has been considered by several authors, mainly Miles, who has taken it as the starting point of a series of papers which are the IMSÍS of the so-called stochastic geometry