Steinmetz Solid
The solid common to two (or three) right circular Cylinders of equal Radiiintersecting at Right Angles is called the Steinmetz solid. (Two Cylindersintersecting at Right Angles are sometimes called a Bicylinder, and three intersectingCylinders a Tricylinder.)
The Volume common to two intersecting right Cylinders of Radius 
is
 | (1) |
and 
with 
, thenthe Volume common to them is | (2) |
is the complete Elliptic Integral of the First Kind, 
is the completeElliptic Integral of the Second Kind, and 
is the Modulus.The curves of intersection of two cylinders of Radii and , shown above, are given by the parametric equations
 |  |  | (3) |
 | |  | (4) |
 | |  | (5) |
(Gray 1993).
The Volume common to two Elliptic Cylinders
 | (6) |
is | (7) |
(Bowman 1961, p. 34).For three Cylinders of Radii intersecting at Right Angles, the Volume of intersection is
 | (8) |
References
Bowman, F. Introduction to Elliptic Functions, with Applications. New York: Dover, 1961. Gardner, M. The Unexpected Hanging and Other Mathematical Diversions. Chicago, IL: Chicago University Press, pp. 183-185, 1991. Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 149-150, 1993. Wells, D. G. #555 in The Penguin Book of Curious and Interesting Puzzles. London: Penguin Books, 1992.
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