Spherical cap

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The spherical cap is the purple section.

In geometry, a spherical cap is a portion of a sphere cut off by a plane. If the plane passes through the center of the sphere, so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere.

If the radius of the sphere is called r, the radius of the base of the cap called a, and the height of the cap called h, the volume of the spherical cap is then

V = πh(3a2 + h2) / 6

and the curved surface area of the spherical cap is

A = 2πrh.

Note also that in the upper hemisphere of the diagram, \scriptstyle h = r - \sqrt{r^2 - a^2}, and in the lower hemisphere \scriptstyle h = r + \sqrt{r^2 - a^2}; hence in either hemisphere \scriptstyle a = \sqrt{h(2r-h)} and so an alternative expression for the volume is

V = πh2(3r − h) / 3.

[edit] Hyperspherical cap

Generally, the n-dimensional volume of a hyperspherical cap of height h and radius r in n-dimensional Euclidean space is given by

V = \frac{\pi ^ {\frac{n-1}{2}}\, r^{n}}{\,\Gamma \left ( \frac{n+1}{2} \right )} \int\limits_{0}^{\cos^{-1}\left(\frac{r-h}{r}\right)}\sin^n (t) \,\mathrm{d}t

where Γ (the gamma function) is given by \Gamma(z) = \int_0^\infty t^{z-1} \mathrm{e}^{-t}\,\mathrm{d}t .

The formula for V can be expressed in terms of the volume of the unit n-ball C_{n}={\scriptstyle \pi^{n/2}/\Gamma[1+\frac{n}{2}]} and the hypergeometric series 2F1 as

V = C_{n} \, r^{n} \left( \frac{1}{2}\, - \,\frac{r-h}{r} \,\frac{\Gamma[1+\frac{n}{2}]}{\sqrt{\pi}\,\Gamma[\frac{n+1}{2}]} {\,\,}_{2}F_{1}\left(\tfrac{1}{2},\tfrac{1-n}{2};\tfrac{3}{2};\left(\tfrac{r-h}{r}\right)^{2}\right)\right) ,

where \scriptstyle 0\le h\le r.

[edit] See also

[edit] External links

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