Buffon-Laplace Needle Problem as a geometric probabilistic approach to filtration process

Buffon-Laplace Needle Problem considers a needle of a length $l$ randomly dropped on a large plane distributed with vertically parallel lines with distances $a$ and $b$ ($a \geqslant b$), respectively. As a classical problem in stochastic probability, it serves as a mathematical basis of various physical literature, such as the efficiency of a filter and the emergence of clogging in filtration process. Yet its potential application is limited by previous focus on its original form of the `short' needle case of $l < b$ and its analytical difficulty in a general sense. Here, rather than a `short' needle embedded in two-dimensional space, we analytically solve problem versions with needles and spherocylinders of arbitrary length and radius embedded in two- and three-dimensional spaces dropped on a grid with any rectangular shape. We further confirm our analytical theory with Monte Carlo simulation. Our framework here helps to provide a geometric analytical perspective to filtration process, and also extend the analytical power of the needle problem into unexplored parameter regions for physical problems involving stochastic processes.