Cylinder-Cylinder Intersection

Cylinder-Cylinder Intersection

Consider two cylinders as illustrated above (Hubbell 1965) where the cylinders have radii r_1 and r_2 with r_1<=r_2, the larger cylinder is oriented along the z-axis, and where the axes of the two cylinders intersecting at an angle beta. Then the volume of the region of intersection is given by

=(4pir_2^3)/(sinbeta)sum_(k=1)^(infty)(1/2; n)(1/2; n-1)k^(2n),



Here, K(k) and E(k) are complete elliptic integrals of the first and second kinds, respectively, _2F_1(a,b;c;z) is a hypergeometric function, and (n; k) 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.

See also

Cylinder, Steinmetz Solid

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Hubbell, J. H. "Common Volume of Two Intersecting Cylinders." J. Research National Bureau of Standards--C. Engineering and Instrumentation 69C, 139-143, April-June 1965.

Cite this as:

Weisstein, Eric W. "Cylinder-Cylinder Intersection." From MathWorld--A Wolfram Web Resource.

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