This may be a question for a physics forum, but I think the meat of the problem is probably better suited for math people, so forgive me if this isn't the right place to ask this question.
Essentially, I have this image of an array of parallel finite lines
Array of parallel lines with perpendicular intersecting line
and another line drawn through it with intersections marked with red asterisks. A single line drawn on a page should be characterized by 4 numbers: its x and y positions, its orientation, and its length. An array of these lines as shown should have 4 equivalent distribution: x and y distributions in space, length distribution, and orientation distribution.
The image above is a simulation of lines with a random x and y positions (equal probability of being generated anywhere), all the same orientation, and a variable length distribution, a sample histogram of which is given below on the left
Length distribution on the left, intersection distribution on the left
and on the right is a distribution made from the distance between intersections of the fist image.
My question is this: given these four distributions, what is the expected distribution of intersection distances as in the image above on the right? Now, my intuition says that all 4 distributions (x distribution, y distribution, orientation distribution, and length distribution) should impact the intersection distance distribution but I'm not sure.
Hopefully this was clear. If you don't know how to do this, I would appreciate being pointed in the right direction to figure this out myself. Thanks!