Consider two cylinders as illustrated above (Hubbell 1965) where the cylinders have radii and with , the larger cylinder is oriented along the -axis, and where the axes of the two cylinders intersecting at an angle . Then the volume of the region of intersection is given by
(1)
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(2)
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(3)
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(4)
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(5)
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where
(6)
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Here, and are complete elliptic integrals of the first and second kinds, respectively, is a hypergeometric function, and is a binomial coefficient.
The intersection of two (or three) right cylinders of equal radii intersecting at right angles is known as the Steinmetz solid.