# Is there a closed form expression for the common volume of two intersecting cylinders with finite length?

There is an article I'm looking at (http://nvlpubs.nist.gov/nistpubs/jres/69C/jresv69Cn2p139_A1b.pdf) that discusses the common volume between two intersecting cylinders, intersecting at an angle. The problem I'm working on has two intersecting cylinders of the same radii. So, my solution should be of the form $$V(r,\beta) = \frac{16 r^3}{3\sin\beta}.$$ The question I have is how can you change this to account for non-infinite intersecting cylinders? That is, what is the change needed so that $$\lim_{\beta\to0}\frac{16 r^3}{3\sin\beta}=\pi r^2h.$$

EDIT: It can also be assumed that the cylinders of equal height and radius intersect at their mid-point.

• I assume the axes of the two cylinders intersect? – user7530 Apr 10 '17 at 22:17
• They are. It's like the picture in that article, except the cylinders have the a finite and equal height. – Lou Apr 10 '17 at 22:18
• Just write the general volume integral, if it has closed form then the answer is yes. – Arjang Apr 10 '17 at 22:20
• This problem sounds extremely challenging. Depending on your application you may be better off approximating the volume of overlap by discretizing the cylinders into e.g. $n$-gonal prisms – user7530 Apr 10 '17 at 23:01
• As a very general variation (finite length, non-orthogonal), this post barely has any content. Thus I'd like to link it to the most basic version listed in the meta post for (abstract) duplicates. – Lee David Chung Lin Jan 22 at 12:11