A cylinder-cylinder intersection is the common region of two cylinders. For cylinders with radii 
whose axes intersect at an angle 
,
as illustrated above (Hubbell 1965), the volume of the
intersection is given by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  | ![(8r_2^3)/(3sinbeta)[(1+k^2)E(k)-(1-k^2)K(k)]](/images/equations/Cylinder-CylinderIntersection/Inline11.svg) |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
where
 |
(6)
|
Here, 
and 
are complete elliptic integrals
of the first and second
kinds, respectively, 
is a hypergeometric
function, and 
is a binomial coefficient.
The elliptic expression is interpreted by continuity
when 
.
The intersection of two (or three) right cylinders of equal radii intersecting at right angles is known as the Steinmetz solid.
