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Cylinder-Sphere Intersection


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

CylinderSpherePOV
CylinderSphereMathematica

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.

CylinderSphereCurve
CylinderSphereSurface

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,

x=+/-sqrt(2(R-r)(R-z))
(3)
y=+/-sqrt((R-z)(2r-R+z)).
(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 z is given by

ds=sqrt(1+((dy)/(dz))^2)dz
(5)
=r/(sqrt((R-z)(2r-R+z)))dz.
(6)

The surface area of one quarter of the surface is then

S_(1/4)=intx(z)ds
(7)
=int_(R-2r)^Rsqrt(2(R-r)(R-z))(rdz)/(sqrt((R-z)(2r-R-z)))
(8)
=int_(R-2r)^Rrsqrt((2(R-r))/(2r-R+z))dz,
(9)

where some care is needed treating the lower limit,

S_(1/4)=lim_(r^'->r^-)4r[sqrt(r(R-r))-sqrt((R-r)(r-r^'))]
(10)
=4r^(3/2)sqrt(R-r).
(11)

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 Curve

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References

Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85-91, May 1998.

Referenced on Wolfram|Alpha

Cylinder-Sphere Intersection

Cite this as:

Weisstein, Eric W. "Cylinder-Sphere Intersection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Cylinder-SphereIntersection.html

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