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 r internally tangent to a sphere of radius R was given in a Sangaku problem from 1825.
The easiest way to determine the solution is to solve the simultaneous equations
| x^2+y^2+z^2=R^2 |
(1)
|
| y^2+[z-(R-r)]^2=r^2 |
(2)
|
for x and y,
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 z is given by
The surface area of one quarter of the surface is then
where some care is needed treating the lower limit,
The total surface area is then
| S=4S_(1/4)=16r^(3/2)sqrt(R-r), |
(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.)
See also
Cylinder, Sphere, Sphere-Sphere Intersection, Viviani's CurveExplore with Wolfram|Alpha
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References
Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85-91, May 1998.Referenced on Wolfram|Alpha
Cylinder-Sphere IntersectionCite this as:
Weisstein, Eric W. "Cylinder-Sphere Intersection." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Cylinder-SphereIntersection.html