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I need to find the volume of an arbitrary intersection of a cube and a cylinder. the sides of the cube ($a$) will always be less than the diameter of the cylinder, such that a cube can fit fully inside the cylinder. Anyone have any idea how to find the volume of the intersection with the cube at an arbitrary position?

I found this case which seems to be the right shape for the intersection but the solution stipulates that $a>r$ which will never be true in my case.

It is also possible to assume that the intersection is close to a wedge (ie $a<<r$) but this will not always be the case.

Note: You can assume the cylinder is infinite.

Raab70
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  • Is it really worth the hassle to come up with analytic solutions instead a numerical solution? – mvw Oct 06 '14 at 14:23
  • It has become necessary. I have been using a numerical solution to simulate ~900. I now need to simulate ~4 million cases. – Raab70 Oct 06 '14 at 14:27
  • Any restrictions on the rotational degrees of freedom of the cube? – mvw Oct 06 '14 at 14:32
  • Preferably no, but a "simple" case of when the cube is parallel to the cylinder would also be helpful. – Raab70 Oct 06 '14 at 14:35
  • the sides of the cube will always be less than the diameter of the cylinder, such that a cube can fit fully inside the cylinder - This sentence is wrong. Perhaps you meant diagonals ? – Lucian Oct 06 '14 at 15:31
  • Similar question, to draw inspiration from. – Lucian Oct 06 '14 at 15:33

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Citing from an answer to the sub case area of intersection of circle and square:

The formula will be a mess, no matter how you look at it. There will be a lot of cases, period.

I would strongly advise to apply numerical methods here. Or maybe calculating some points in your configuration space (which might be the set of all $(a/r, x, y, \alpha, \beta))$ and then resort to approximation.

Community
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mvw
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