Geometric Probabilities for an Arbitrary Convex Body of Revolution in E3 and Certain Lattices

Problems of geometric probability for an arbitrary convex body of resolution in the euclidean space E3 has been investigated in [1]. In [9] Buffon’s problem is solved for a lattice of right-angled parallelepipeds in the 3-dimensional space. In this note we want to use the results in [5] for to solve problems of intersection for a particular lattice that we describe: the fundamental cell C0 of the lattice R is a right-angled prism of height c and whose basis is the following: Let K be an arbitrary convex body of resolution with centroid G and oriented axis of rotation r. The line r is determined by the angle θ between r and the z-axis and by the angle φ between the projection of r on the xy-plane and the x-axis. Hence r = r(θ, φ). Then the length L of the projection of K on the z-axis is given by