Spherical Cap - Hyperspherical Cap

Hyperspherical Cap

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

where (the gamma function) is given by .

The formula for can be expressed in terms of the volume of the unit n-ball and the hypergeometric function or the regularized incomplete beta function as

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

and the area formula can be expressed in terms of the area of the unit n-ball as

,

where .

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