# Spherical cap

An example of a spherical cap in blue (and another in red.)

In geometry, a spherical cap, spherical dome, or spherical segment of one base 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.

## Volume and surface area

If the radius of the base of the cap is ${\displaystyle a}$, and the height of the cap is ${\displaystyle h}$, then the volume of the spherical cap is[1]

${\displaystyle V={\frac {1}{6}}\pi h(3a^{2}+h^{2})}$

and the curved surface area of the spherical cap is[1]

${\displaystyle A=2\pi rh}$

or

${\displaystyle A=2\pi r^{2}(1-\cos \theta )}$

The relationship between ${\displaystyle h}$ and ${\displaystyle r}$ is irrelevant as long as ${\displaystyle 0\leq h\leq 2r}$. The red section of the illustration is also a spherical cap.

The parameters ${\displaystyle a}$ , ${\displaystyle h}$ and ${\displaystyle r}$ are not independent:

${\displaystyle r^{2}=(r-h)^{2}+a^{2}=r^{2}+h^{2}-2rh+a^{2}}$
${\displaystyle r={\frac {a^{2}+h^{2}}{2h}}}$

Substituting this into the area formula gives:

${\displaystyle A=2\pi {\frac {(a^{2}+h^{2})}{2h}}h=\pi (a^{2}+h^{2})}$.

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

${\displaystyle V={\frac {\pi h^{2}}{3}}(3r-h)}$.

The volume may also be found by integrating under a surface of rotation and factorizing as follows.

${\displaystyle V=\int _{r\cos \theta }^{r}\pi \left(r^{2}-x^{2}\right)dx={\frac {\pi }{3}}r^{3}(2-3\cos \theta +\cos ^{3}\theta )={\frac {\pi }{3}}r^{3}(2+\cos \theta )(1-\cos \theta )^{2}}$.

## Applications

### Volumes of union and intersection of two intersecting spheres

The volume of the union of two intersecting spheres of radii ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ is [2]

${\displaystyle V=V^{(1)}-V^{(2)}}$,

where

${\displaystyle V^{(1)}={\frac {4\pi }{3}}r_{1}^{3}+{\frac {4\pi }{3}}r_{2}^{3}}$

is the sum of the volumes of the two isolated spheres, and

${\displaystyle V^{(2)}={\frac {\pi h_{1}^{2}}{3}}(3r_{1}-h_{1})+{\frac {\pi h_{2}^{2}}{3}}(3r_{2}-h_{2})}$

the sum of the volumes of the two spherical caps forming their intersection. If ${\displaystyle d is the distance between the two sphere centers, elimination of the variables ${\displaystyle h_{1}}$ and ${\displaystyle h_{2}}$ leads to[3][4]

${\displaystyle V^{(2)}={\frac {\pi }{12d}}(r_{1}+r_{2}-d)^{2}\left(d^{2}+2d(r_{1}+r_{2})-3(r_{1}-r_{2})^{2}\right)}$ .

### Surface area bounded by circles of latitude

The surface area bounded by two circles of latitude is the difference of surface areas of their respective spherical caps. For a sphere of radius ${\displaystyle r}$, and latitudes ${\displaystyle \phi _{1}}$ and ${\displaystyle \phi _{2}}$, the area is [5]

${\displaystyle A=2\pi r^{2}|\sin \phi _{1}-\sin \phi _{2}|}$

For example, assuming the Earth is a sphere of radius 6371 km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016[6]) is 2π·6371²|sin 90° − sin 66.56°| = 21.04 million km², or 0.5·|sin 90° − sin 66.56°| = 4.125% of the total surface area of the Earth.

## Generalizations

### Sections of other solids

The spheroidal dome is obtained by sectioning off a portion of a spheroid so that the resulting dome is circularly symmetric (having an axis of rotation), and likewise the ellipsoidal dome is derived from the ellipsoid.

### Hyperspherical cap

Generally, the ${\displaystyle n}$-dimensional volume of a hyperspherical cap of height ${\displaystyle h}$ and radius ${\displaystyle r}$ in ${\displaystyle n}$-dimensional Euclidean space is given by [7]

${\displaystyle V={\frac {\pi ^{\frac {n-1}{2}}\,r^{n}}{\,\Gamma \left({\frac {n+1}{2}}\right)}}\int \limits _{0}^{\arccos \left({\frac {r-h}{r}}\right)}\sin ^{n}(t)\,\mathrm {d} t}$

where ${\displaystyle \Gamma }$ (the gamma function) is given by ${\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}\mathrm {e} ^{-t}\,\mathrm {d} t}$.

The formula for ${\displaystyle V}$ can be expressed in terms of the volume of the unit n-ball ${\displaystyle C_{n}={\scriptstyle \pi ^{n/2}/\Gamma [1+{\frac {n}{2}}]}}$ and the hypergeometric function ${\displaystyle {}_{2}F_{1}}$ or the regularized incomplete beta function ${\displaystyle I_{x}(a,b)}$ as

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

and the area formula ${\displaystyle A}$ can be expressed in terms of the area of the unit n-ball ${\displaystyle A_{n}={\scriptstyle 2\pi ^{n/2}/\Gamma [{\frac {n}{2}}]}}$ as

${\displaystyle A={\frac {1}{2}}A_{n}\,r^{n-1}I_{(2rh-h^{2})/r^{2}}\left({\frac {n-1}{2}},{\frac {1}{2}}\right)}$ ,

where ${\displaystyle \scriptstyle 0\leq h\leq r}$.

Earlier in [8] (1986, USSR Academ. Press) the following formulas were derived: ${\displaystyle A=A_{n}p_{n-2}(q),V=C_{n}p_{n}(q)}$, where ${\displaystyle q=1-h/r(0\leq q\leq 1),p_{n}(q)=(1-G_{n}(q)/G_{n}(1))/2}$,

${\displaystyle G_{n}(q)=\int \limits _{0}^{q}(1-t^{2})^{(n-1)/2}dt}$.

For odd ${\displaystyle n=2k+1:}$

${\displaystyle G_{n}(q)=\sum _{i=0}^{k}(-1)^{i}{\binom {k}{i}}{\frac {q^{2i+1}}{2i+1}}}$.

#### Asymptotics

It is shown in [9] that, if ${\displaystyle n\to \infty }$ and ${\displaystyle q{\sqrt {n}}={\text{const.}}}$, then ${\displaystyle p_{n}(q)\to 1-F({q{\sqrt {n}}})}$ where ${\displaystyle F()}$ is the integral of the standard normal distribution.

A more quantitive way of writing this, is in [10] where the bound ${\displaystyle A/A_{n}=n^{\Theta (1)}\cdot [(2-h/r)h/r]^{n/2}}$ is given. For large caps (that is when ${\displaystyle (1-h/r)^{4}\cdot n=O(1)}$ as ${\displaystyle n\to \infty }$), the bound simplifies to ${\displaystyle n^{\Theta (1)}\cdot e^{-(1-h/r)^{2}n/2}}$.

## References

1. ^ a b Polyanin, Andrei D; Manzhirov, Alexander V. (2006), Handbook of Mathematics for Engineers and Scientists, CRC Press, p. 69, ISBN 9781584885023.
2. ^ Connolly, Michael L. (1985). "Computation of molecular volume". J. Am. Chem. Soc. 107: 1118–1124. doi:10.1021/ja00291a006.
3. ^ Pavani, R.; Ranghino, G. (1982). "A method to compute the volume of a molecule". Comput. Chem. 6: 133–135. doi:10.1016/0097-8485(82)80006-5.
4. ^ Bondi, A. (1964). "Van der Waals volumes and radii". J. Phys. Chem. 68: 441–451. doi:10.1021/j100785a001.
5. ^ Scott E. Donaldson, Stanley G. Siegel. "Successful Software Development". Retrieved 29 August 2016.
6. ^ "Obliquity of the Ecliptic (Eps Mean)". Neoprogrammics.com. Retrieved 2014-05-13.
7. ^ Li, S (2011). "Concise Formulas for the Area and Volume of a Hyperspherical Cap". Asian J. Math. Stat. 4 (1): 66–70. doi:10.3923/ajms.2011.66.70.
8. ^ Chudnov, Alexander M. (1986). "On minimax signal generation and reception algorithms (rus.)". Problems of Information Transmission. 22 (4): 49–54.
9. ^ Chudnov, Alexander M (1991). "Game-theoretical problems of synthesis of signal generation and reception algorithms (rus.)". Problems of Information Transmission. 27 (3): 57–65.
10. ^ Anja Becker, Léo Ducas, Nicolas Gama, and Thijs Laarhoven. 2016. New directions in nearest neighbor searching with applications to lattice sieving. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms (SODA '16), Robert Kraughgamer (Ed.). Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 10-24.