Buffon type problem in the euclidean spaceR ³

Abstract

Consider a sphere of radiusr whose centre is uniformly distributed inR ³ in the sense that the kinematic measure associated with the centre is that given in Santaló (1976). Consider a lattice of planes whose elementary cell consists of a prism of heighth and with the base an arbitrary triangle of sidesa, b, c. Bosetto (1997) considered the case when the base of the prism is a right-angled triangle, computed the probability that the random sphere cuts at least one of the planes of the lattice and established some independence properties of certin events. Here the same probability is computed for prisms of arbitrary triangular bases and expressed in terms of symmetric expressions. Independence properties and generalization to prisms with arbitrary polygonal bases are also considered.