Please consider the case where I have 'N' rods of length L (and width W) placed on a one- or two-dimensional surface with dimensions [0, A] in 1D, and [ [0, A], [0, B] ] in 2D. For the two-dimensional case, L and W are << A or B.

As a function of the number of rods N, and the relative dimensions of the rods and the surface on which they are placed, is there a reasonably easy derivation for the number of expected intersections between rods / the probability of an intersection occurring? I feel like this should have been solved somewhere in the literature, but I was unable to find anything.

Edit - When I mentioned the rods should be 'placed', I failed to clarify that my treatment has been to assume that one end of each rod is placed with uniform random probability somewhere inside the specified dimensions of the one- or two-dimensional surface, the angle of the rod should be random, and any rod sections outside the bounded surface should either be treated or ignored depending on how easy it makes treating boundary conditions.

percolationof randomly placed rods often goes by the name "stick percolation" in the literature. See for instance this paper of Rahul Roy for some rigorous work math.bme.hu/~balint/oktatas/perkolacio/percolation_papers/… , and this paper of Jiantong Li and Shi-Li Zhang for some numerical work link.aps.org/doi/10.1103/PhysRevE.80.040104 – j.c. Apr 12 '11 at 16:49