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 sphere is $r$ , and the height of the cap is $h$ , then the volume of the spherical cap is^[1]

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

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

A=2\pi rh

or

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

in which $\theta$ is the polar angle between the rays from the center of the sphere to the apex of the cap (the pole) and the edge of the disk forming the base of the cap. (If it $\phi$ denotes the latitude in geographic coordinates, $\theta +\phi =\pi /2=90^{\circ }\,$ .)

The relationship between $h$ and $r$ is irrelevant as long as $0\leq h\leq 2r$ . For example, the red section of the illustration is also a spherical cap for which $h>r$ .

These formulas can be rewritten to use the radius $a$ of the base of the cap instead of $r$ , using the relationship:

r^{2}=(r-h)^{2}+a^{2}=r^{2}+h^{2}-2rh+a^{2}\,,

so that

r={\frac {a^{2}+h^{2}}{2h}}\,.

Substituting this into the formulas gives:

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

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

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

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 $r_{1}$ and $r_{2}$ is ^[2]

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

where

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

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 $d\leq r_{1}+r_{2}$ is the distance between the two sphere centers, elimination of the variables $h_{1}$ and $h_{2}$ leads to^[3]^[4]

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 parallel disks

The curved surface area of the spherical segment bounded by two parallel disks is the difference of surface areas of their respective spherical caps. For a sphere of radius $r$ , and caps with heights $h_{1}$ and $h_{2}$ , the area is

A=2\pi r|h_{1}-h_{2}|\,,

or, using geographic coordinates with latitudes $\phi _{1}$ and $\phi _{2}$ ,^[5]

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 2 $π$ ·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.

This formula can also be used to demonstrate that half the surface area of the Earth lies between latitudes 30° South and 30° North in a spherical zone which encompasses all of the Tropics.

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 $n$ -dimensional volume of a hyperspherical cap of height $h$ and radius $r$ in $n$ -dimensional Euclidean space is given by ^[7]

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 $\Gamma$ (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 function ${}_{2}F_{1}$ or the regularized incomplete beta function $I_{x}(a,b)$ 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)={\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 $A$ can be expressed in terms of the area of the unit n-ball $A_{n}={\scriptstyle 2\pi ^{n/2}/\Gamma [{\frac {n}{2}}]}$ as

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 $\scriptstyle 0\leq h\leq r$ .

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

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

For odd $n=2k+1:$

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

Asymptotics

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

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

References

1 2 Polyanin, Andrei D; Manzhirov, Alexander V. (2006), Handbook of Mathematics for Engineers and Scientists, CRC Press, p. 69, ISBN 9781584885023 .
↑ Connolly, Michael L. (1985). "Computation of molecular volume". J. Am. Chem. Soc. 107: 1118–1124. doi:10.1021/ja00291a006.
↑ 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.
↑ Bondi, A. (1964). "Van der Waals volumes and radii". J. Phys. Chem. 68: 441–451. doi:10.1021/j100785a001.
↑ Scott E. Donaldson, Stanley G. Siegel. "Successful Software Development". Retrieved 29 August 2016.
↑ "Obliquity of the Ecliptic (Eps Mean)". Neoprogrammics.com. Retrieved 2014-05-13.
↑ 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.
↑ Chudnov, Alexander M. (1986). "On minimax signal generation and reception algorithms (rus.)". Problems of Information Transmission. 22 (4): 49–54.
↑ Chudnov, Alexander M (1991). "Game-theoretical problems of synthesis of signal generation and reception algorithms (rus.)". Problems of Information Transmission. 27 (3): 57–65.
↑ 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.

External links

Weisstein, Eric W. "Spherical cap". MathWorld. Derivation and some additional formulas.
Online calculator for spherical cap volume and area.
Summary of spherical formulas.

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[handbook-1] 1 2 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.

[Minimax-86-8] Chudnov, Alexander M. (1986). "On minimax signal generation and reception algorithms (rus.)". Problems of Information Transmission. 22 (4): 49–54.

[Game-91-9] 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.