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Let the sets of points $(p_1, p_2)$ and $(q_1,q_2)$ define the endpoints of the long axes for two finite cylinders, $C_1$ and $C_2$, of radii $r_1$ and $r_2$, in three-dimensional space. What is a general method of calculating the exact intersection volume of the two cylinders? Does the problem become easier if one only wants to know the volume within some small error?

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I am afraid that the exact formula might be cumbersome to derive. But for approximation, there is a general method to compute volume of any convex body, originating from the works of Dyer--Frieze--Kannan method. From computation point of view these methods are very easy to implement: all one needs is to be able tell if a point is the body, and to simulate certain very simple random walk inside the body. The hard part is to prove that the resulting algorithm works, but it is not needed to run the algorithm. See renyi.hu/~miki/kopenwww.pdf for a survey of these algorithms. –  Boris Bukh Feb 18 '12 at 15:49
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