Chord calculus and stochastic geometry

Abstract

Chord calculus is a collection of integration procedures applied to to the combinatorial decompositions that give the solution of the Buffon-Sylvester problem for n needles in a plane or the similar problem in IR ³. It is a source of various integral geometry identities, some of which find their application in Stochastic geometry. In the present paper these applications are focused on random convex polygons and polyhedrons, where we define certain classes where rather simple tomography analysis is possible. The choice of these classes (the Independent Angles class and the Independent Orientations class) is due to the nature of the results of the Chord calculus. The last section points at an application of the convex polygons from the Independent Angles class to Boolean sets in the plane (Boolean models) whose probability distibutions are invariant with respect to the group of Euclidean motions of the plane.