Laplace type problems for a triangular lattice and different body test

Hitting probabilities for non-convex lattice

“…As in the previous paragraphs, we compute the probability P (3) int that a segment s intersects a side of fundamental cell C 3.…”

A Laplace problem for a regular lattice with symmetric and asymmetric pentagonal cell and different obstacles

“…Considering a segment s of random position and of constant lenght l with l < min (a, b, c), we want to compute the probabiltiy that this segment intersects a side of lattice, that is, the probability P (2) int that the segment s intersects a side of the fundamental cell. The position of the segment s is determinated by the middle point O and by the angle ϕ that the segment forms with the axis x.…”

“…The position of the segment s is determinated by the middle point O and by the angle ϕ that the segment forms with the axis x. We consider the limit positions of the segment s and let C (2) 0 (ϕ) be determinated figure from these positions for a prefixed value of ϕ, (see fig. 4):…”

A Laplace type problem for a convex-concave lattice and non-uniform distribution

“…By starting with the results obtained by Poincaré [10] and Stoka [11], in the last years, many authors have managed to carry out in-depth studies about an extremely large variety of Buffon-Laplace problems. They considered different types of fundamental cells and studied geometric probability problems, both classic and with obstacles (see [1] - [8]). In 2016, Stoka [9] presented a classic Buffon-Laplace type problem by considering as the test body a random segment with a non-uniform distribution.…”