  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'
two-dimensional needle, we analytically solve the problems with two- and
three-dimensional needles and spherocylinders of arbitrary length and radius
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.
