Cylinder-Sphere Intersection

The curve formed by the intersection of a cylinder and a sphere is known as Viviani's curve.

The problem of finding the lateral surface area of a cylinder of radius internally tangent to a sphere of radius was given in a Sangaku problem from 1825.

The easiest way to determine the solution is to solve the simultaneous equations

(1)

(2)

for and ,

			(3)
			(4)

These give the parametric equations for Viviani's curve in this case (left figure). The surface area can then be found by constructing a series of curved segments (right figure). The arc length element around the surface of the cylinder at a height is given by

			(5)
			(6)

The surface area of one quarter of the surface is then

			(7)
			(8)
			(9)

where some care is needed treating the lower limit,

			(10)
			(11)

The total surface area is then

(12)

a result obtained in a more roundabout geometric arguments by Rothman (1998). (Note that the answer printed in the original Rothman article was incorrect; the corrected answer has been posted on the Internet version of the article.)