Spherical Cap
A spherical cap is the region of a sphere which lies above (or below) a given plane. If the plane passes through the center of the sphere, the cap is a called a hemisphere, and if the cap is cut by a second plane, the spherical frustum is called a spherical segment. However, Harris and Stocker (1998) use the term "spherical segment" as a synonym for what is here called a spherical cap and "zone" for spherical segment.
Let the sphere have radius R, then the volume of a spherical cap of height h and base radius a is given by the equation of a spherical segment
| V_(spherical segment)=1/6pih(3a^2+3b^2+h^2) |
(1)
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with b=0, giving
| V_(cap)=1/6pih(3a^2+h^2). |
(2)
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Using the Pythagorean theorem gives
| (R-h)^2+a^2=R^2, |
(3)
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which can be solved for a^2 as
| a^2=2Rh-h^2, |
(4)
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so the radius of the base circle is
| a=sqrt(h(2R-h)), |
(5)
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and plugging this in gives the equivalent formula
| V_(cap)=1/3pih^2(3R-h). |
(6)
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In terms of the so-called contact angle (the angle between the normal to the sphere at the bottom of the cap and the base plane)
| R-h=Rsinalpha |
(7)
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| [画像: alpha=sin^(-1)((R-h)/R), ] |
(8)
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so
| V_(cap)=1/3piR^3(2-3sinalpha+sin^3alpha). |
(9)
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The geometric centroid occurs at a distance
| [画像: z^_=(3(2R-h)^2)/(4(3R-h)) ] |
(10)
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above the center of the sphere (Harris and Stocker 1998, p. 107).
The cap height h at which the spherical cap has volume equal to half a hemisphere is given by
| h_(1/2)=1-2cos(4/9pi). |
(11)
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Consider a cylindrical box enclosing the cap so that the top of the box is tangent to the top of the sphere. Then the enclosing box has volume
so the hollow volume between the cap and box is given by
| V_(box)-V_(cap)=1/3piR^3(1-3sin^2alpha+2sin^3alpha). |
(15)
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The surface area of the spherical cap is given by the same equation as for a general zone:
See also
Contact Angle, Frustum, Hemisphere, Solid of Revolution, Sphere, Spherical Ring, Spherical Segment, Spherical Wedge, Surface of Revolution, Torispherical Dome, ZoneExplore with Wolfram|Alpha
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References
Harris, J. W. and Stocker, H. "Spherical Segment (Spherical Cap)." §4.8.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 107, 1998.Kern, W. F. and Bland, J. R. "Spherical Segment." §36 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 97-102, 1948.Referenced on Wolfram|Alpha
Spherical CapCite this as:
Weisstein, Eric W. "Spherical Cap." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SphericalCap.html