spherical cap

The volume of the spherical cap and the area of the curved surface may be calculated using combinations of

• The radius $r$ of the sphere
• The radius $a$ of the base of the cap
• The height $h$ of the cap
• The polar angle $\backslash theta$ 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.

These variables are inter-related through the formulas

$a\; =\; r\; \backslash sin\; \backslash theta$, $h\; =\; r\; (\; 1\; -\; \backslash cos\; \backslash theta\; )$, $2hr\; =\; a^2\; +\; h^2$,

and $2\; h\; a\; =\; (a^2\; +\; h^2)\backslash sin\; \backslash theta$.

If $\backslash phi$ denotes the latitude in geographic coordinates, then $\backslash theta+\backslash phi\; =\; \backslash pi/2\; =\; 90^\backslash circ\backslash ,$, and $\backslash cos\; \backslash theta\; =\; \backslash sin\; \backslash phi$.

Note that aside from the calculus based argument below, the area of the spherical cap may be derived from the volume $V\_\{sec\}$ of the spherical sector, by an intuitive argument,{{cite web |last1=Shekhtman |first1=Zor |title=Unizor - Geometry3D - Spherical Sectors |url=https://www.youtube.com/watch?v=ts3J5onzvQg&t=8m54s |archive-url=https://ghostarchive.org/varchive/youtube/20211222/ts3J5onzvQg |archive-date=2021-12-22 |url-status=live|website=YouTube |publisher=Zor Shekhtman |access-date=31 Dec 2018}}{{cbignore}} as

:$A\; =\; \backslash frac\{3\}\{r\}V\_\{sec\}\; =\; \backslash frac\{3\}\{r\}\; \backslash frac\{2\backslash pi\; r^2h\}\{3\}\; =\; 2\backslash pi\; rh\backslash ,.$

The intuitive argument is based upon summing the total sector volume from that of infinitesimal triangular pyramids. Utilizing the pyramid (or cone) volume formula of $V\; =\; \backslash frac\{1\}\{3\}\; bh\text{'}$, where $b$ is the infinitesimal area of each pyramidal base (located on the surface of the sphere) and $h\text{'}$ is the height of each pyramid from its base to its apex (at the center of the sphere). Since each $h\text{'}$, in the limit, is constant and equivalent to the radius $r$ of the sphere, the sum of the infinitesimal pyramidal bases would equal the area of the spherical sector, and:

:$V\_\{sec\}\; =\; \backslash sum\{V\}\; =\; \backslash sum\backslash frac\{1\}\{3\}\; bh\text{'}\; =\; \backslash sum\backslash frac\{1\}\{3\}\; br\; =\; \backslash frac\{r\}\{3\}\; \backslash sum\; b\; =\; \backslash frac\{r\}\{3\}\; A$

File:Spherical cap from rotation.svg

The volume and area formulas may be derived by examining the rotation of the function

:$f(x)=\backslash sqrt\{r^2-(x-r)^2\}=\backslash sqrt\{2rx-x^2\}$

for $x\; \backslash in\; [0,h]$, using the formulas the surface of the rotation for the area and the solid of the revolution for the volume.

The area is

:$A\; =\; 2\backslash pi\backslash int\_0^h\; f(x)\; \backslash sqrt\{1+f\text{'}(x)^2\}\; \backslash ,dx$

The derivative of $f$ is

:$f\text{'}(x)\; =\; \backslash frac\{r-x\}\{\backslash sqrt\{2rx-x^2\}\}$

and hence

:$1+f\text{'}(x)^2\; =\; \backslash frac\{r^2\}\{2rx-x^2\}$

The formula for the area is therefore

:$A\; =\; 2\backslash pi\backslash int\_0^h\; \backslash sqrt\{2rx-x^2\}\; \backslash sqrt\{\backslash frac\{r^2\}\{2rx-x^2\}\}\; \backslash ,dx$

= 2\pi \int_0^h r\,dx

= 2\pi r \left[x\right]_0^h

= 2 \pi r h

The volume is

:$V\; =\; \backslash pi\; \backslash int\_0^h\; f(x)^2\; \backslash ,dx$

= \pi \int_0^h (2rx-x^2) \,dx

= \pi \left[rx^2-\frac13x^3\right]_0^h

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

The moment of Inertias of a spherical cap (where the z-axis is the symmetrical axis) about the principle axes (center) of the sphere are:

$J\_\{zz,\backslash text\{cap\}\}\; =\; \backslash frac\{m\; h\; \backslash left(\; 3\; h^2\; -\; 15\; h\; R\; +\; 20\; R^2\; \backslash right)\}\{10\; \backslash left(\; 3\; R\; -\; h\; \backslash right)\}$

$J\_\{xx,\backslash text\{cap\}\}\; =\; J\_\{yy,\; \backslash text\{cap\}\}\; =\backslash frac\{m\; \backslash left(\; -9\; h^3\; +\; 45\; h^2\; R\; -\; 80\; h\; R^2\; +\; 60\; R^3\; \backslash right)\}\{20\; \backslash left(\; 3\; R\; -\; h\; \backslash right)\}$

where m is the mass of the spherical cap, R is the radius of the entire sphere, and h is the height of the spherical cap.

The volume of the union of two intersecting spheres

of radii $r\_1$ and $r\_2$ is

{{cite journal|first1=Michael L.|last1=Connolly|year=1985|doi=10.1021/ja00291a006|title=Computation of molecular volume|journal= Journal of the American Chemical Society|pages=1118–1124|volume=107|issue=5}}

:$V\; =\; V^\{(1)\}-V^\{(2)\}\backslash ,,$

where

:$V^\{(1)\}\; =\; \backslash frac\{4\backslash pi\}\{3\}r\_1^3\; +\backslash frac\{4\backslash pi\}\{3\}r\_2^3$

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

:$V^\{(2)\}\; =\; \backslash frac\{\backslash pi\; h\_1^2\}\{3\}(3r\_1-h\_1)+\backslash frac\{\backslash pi\; h\_2^2\}\{3\}(3r\_2-h\_2)$

the sum of the volumes of the two spherical caps forming their intersection. If $d\; \backslash le\; r\_1+r\_2$ is the

distance between the two sphere centers, elimination of the variables $h\_1$ and $h\_2$ leads

to{{cite journal|doi=10.1016/0097-8485(82)80006-5|year=1982|title=A method to compute the volume of a molecule|journal= Computers & Chemistry|first1=R.|last1=Pavani|first2=G.|last2=Ranghino|volume=6|issue=3|pages=133–135}}{{cite journal|first1=A.|last1=Bondi|doi=10.1021/j100785a001|year=1964|title=Van der Waals volumes and radii|journal= The Journal of Physical Chemistry|volume=68|issue=3|pages=441–451}}

:$V^\{(2)\}\; =\; \backslash frac\{\backslash pi\}\{12d\}(r\_1+r\_2-d)^2\; \backslash left(\; d^2+2d(r\_1+r\_2)-3(r\_1-r\_2)^2\; \backslash right)\backslash ,.$

The volume of a spherical cap with a curved base can be calculated by considering two spheres with radii $r\_1$ and $r\_2$, separated by some distance $d$, and for which their surfaces intersect at $x=h$. That is, the curvature of the base comes from sphere 2. The volume is thus the difference between sphere 2's cap (with height $(r\_2-r\_1)-(d-h)$) and sphere 1's cap (with height $h$),

$\backslash begin\{align\}$

V & = \frac{\pi h^2}{3}(3r_1-h) - \frac{\pi [(r_2-r_1)-(d-h)]^2}{3}[3r_2-((r_2-r_1)-(d-h))]\,, \\

V & = \frac{\pi h^2}{3}(3r_1-h) - \frac{\pi}{3}(d-h)^3\left(\frac{r_2-r_1}{d-h}-1\right)^2\left[\frac{2r_2+r_1}{d-h}+1\right]\,.

\end{align}

This formula is valid only for configurations that satisfy

Consider two intersecting spheres of radii $r\_1$ and $r\_2$, with their centers separated by distance $d$. They intersect if

:$|r\_1-r\_2|\backslash leq\; d\; \backslash leq\; r\_1+r\_2$

From the law of cosines, the polar angle of the spherical cap on the sphere of radius $r\_1$ is

:$\backslash cos\; \backslash theta\; =\; \backslash frac\{r\_1^2-r\_2^2+d^2\}\{2r\_1d\}$

Using this, the surface area of the spherical cap on the sphere of radius $r\_1$ is

:$A\_1\; =\; 2\backslash pi\; r\_1^2\; \backslash left(\; 1+\backslash frac\{r\_2^2-r\_1^2-d^2\}\{2\; r\_1\; d\}\; \backslash right)$

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

:$A=2\; \backslash pi\; r\; |h\_1\; -\; h\_2|\backslash ,,$

or, using geographic coordinates with latitudes $\backslash phi\_1$ and $\backslash phi\_2$,{{cite book|title=Successful Software Development|author=Scott E. Donaldson, Stanley G. Siegel|url=https://books.google.com/books?id=lrix5MNRiu4C&pg=PA354|access-date=29 August 2016|isbn=9780130868268|year=2001}}

:$A=2\; \backslash pi\; r^2\; |\backslash sin\; \backslash phi\_1\; -\; \backslash sin\; \backslash phi\_2|\backslash ,,$

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{{cite web|url=http://www.neoprogrammics.com/obliquity_of_the_ecliptic/ |title=Obliquity of the Ecliptic (Eps Mean) |publisher=Neoprogrammics.com |access-date=2014-05-13}}) is {{math|1= 2π{{thinsp}}⋅{{thinsp}}6371²{{thinsp}}{{abs|sin 90° − sin 66.56°}}}} = {{convert|21.04|e6km2|e6mi2|abbr=unit}}, or {{math|1= 0.5{{thinsp}}⋅{{thinsp}}{{abs|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.

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.

Generally, the $n$-dimensional volume of a hyperspherical cap of height $h$ and radius $r$ in $n$-dimensional Euclidean space is given by:{{cite journal|title=Concise Formulas for the Area and Volume of a Hyperspherical Cap|first1=S.|last1=Li|journal=Asian Journal of Mathematics and Statistics| year=2011| pages=66-70|url=https://docsdrive.com/pdfs/ansinet/ajms/2011/66-70.pdf}}

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

where $\backslash Gamma$ (the gamma function) is given by $\backslash Gamma(z)\; =\; \backslash int\_0^\backslash infty\; t^\{z-1\}\; \backslash mathrm\{e\}^\{-t\}\backslash ,\backslash mathrm\{d\}t$.

The formula for $V$ can be expressed in terms of the volume of the unit n-ball $C\_n\; =\; \backslash pi^\{n/2\}\; /\; \backslash Gamma[1+\backslash frac\{n\}\{2\}]$ and the hypergeometric function $\{\}\_\{2\}F\_\{1\}$ or the regularized incomplete beta function $I\_x(a,b)$ as

$$V\; =\; C\_\{n\}\; \backslash ,\; r^\{n\}\; \backslash left(\; \backslash frac\{1\}\{2\}\backslash ,\; -\; \backslash ,\backslash frac\{r-h\}\{r\}\; \backslash ,\backslash frac\{\backslash Gamma[1+\backslash frac\{n\}\{2\}]\}\{\backslash sqrt\{\backslash pi\}\backslash ,\backslash Gamma[\backslash frac\{n+1\}\{2\}]\}$$

{\,\,}_{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 $A\_\{n\}=\{2\backslash pi^\{n/2\}/\backslash Gamma[\backslash frac\{n\}\{2\}]\}$ as

$$A\; =\backslash frac\{1\}\{2\}A\_\{n\}\; \backslash ,\; r^\{n-1\}\; I\_\{(2rh-h^2)/r^2\}\; \backslash left(\backslash frac\{n-1\}\{2\},\; \backslash frac\{1\}\{2\}\; \backslash right),$$

where $0\backslash le\; h\backslash le\; r$.

A. Chudnov{{cite journal|title=On minimax signal generation and reception algorithms (engl. transl.) | first1=Alexander M.|last1=Chudnov|journal=Problems of Information Transmission| year=1986| volume=22| number=4| pages=49–54|url=https://www.researchgate.net/publication/269008140_Minimax_signal_generation_and_reception_algorithms}} derived the following formulas:

$$A\; =\; A\_n\; r^\{n-1\}\; p\_\; \{\; n-2\; \}\; (q),\backslash ,\; V\; =\; C\_n\; r^\{n\}\; p\_n\; (q)\; ,$$ where

$$q\; =\; 1-h/r\; (0\; \backslash le\; q\; \backslash le\; 1\; ),\; p\_n\; (q)\; =(1-G\_n(q)/G\_n(1))/2\; ,$$

$$G\; \_n(q)=\; \backslash int\; \_0^q\; (1-t^2)\; ^\; \{\; (n-1)\; /2\; \}\; dt\; .$$

For odd $n=2k+1$:

$$G\_n(q)\; =\; \backslash sum\_\{i=0\}^k\; (-1)\; ^i\; \backslash binom\; k\; i\; \backslash frac\; \{q^\{2i+1\}\}\; \{2i+1\}\; .$$

If $n\; \backslash to\; \backslash infty$ and $q\backslash sqrt\; n\; =\; \backslash text\{const.\}$, then $p\_n\; (q)\; \backslash to\; 1-\; F(\{q\; \backslash sqrt\; n\})$ where $F()$ is the integral of the standard normal distribution.{{cite journal|title=Game-theoretical problems of synthesis of signal generation and reception algorithms (engl. transl.)|first1=Alexander M|last1=Chudnov|journal=Problems of Information Transmission | year=1991 | volume=27|number=3|pages=57–65|url=https://www.researchgate.net/publication/268648510_Game-theoretical_problems_of_synthesis_of_signal_generation_and_reception_algorithms}}

A more quantitative bound is $A/(A\_n\; r^\{n-1\})\; =\; n^\{\backslash Theta(1)\}\; \backslash cdot\; [(2-h/r)h/r]^\{n/2\}$.

For large caps (that is when $(1-h/r)^4\backslash cdot\; n\; =\; O(1)$ as $n\backslash to\; \backslash infty$), the bound simplifies to $n^\{\backslash Theta(1)\}\; \backslash cdot\; e^\{-(1-h/r)^2n/2\}$.{{cite conference |last1= Becker |first1= Anja |last2= Ducas |first2= Léo |last3= Gama |first3= Nicolas |last4= Laarhoven |first4= Thijs |date= 10 January 2016 |title= New directions in nearest neighbor searching with applications to lattice sieving |conference= Twenty-seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '16), Arlington, Virginia |editor-last= Krauthgamer |editor-first= Robert |publisher= Society for Industrial and Applied Mathematics |publication-place= Philadelphia |pages= 10–24 |isbn= 978-1-61197-433-1 }}

{{Portal|Maths}}

• Circular segment — the analogous 2D object
• Solid angle — contains formula for n-sphere caps
• Spherical segment
• Spherical sector
• Spherical wedge

{{reflist}}

• {{cite journal|first1= Timothy J. | last1=Richmond |title=Solvent accessible surface area and excluded volume in proteins: Analytical equation for overlapping spheres and implications for the hydrophobic effect |journal= Journal of Molecular Biology|year=1984 | doi=10.1016/0022-2836(84)90231-6 | pmid=6548264 |volume=178 | number=1 |pages=63–89 }}
• {{cite journal| first1=Rolf | last1=Lustig |title=Geometry of four hard fused spheres in an arbitrary spatial configuration |journal= Molecular Physics|year=1986 |volume=59 | number=2 | pages=195–207 |bibcode=1986MolPh..59..195L |doi= 10.1080/00268978600102011}}
• {{cite journal | first1=K. D. | last1=Gibson |first2=Harold A. |last2=Scheraga |title=Volume of the intersection of three spheres of unequal size: a simplified formula |year=1987 | journal= The Journal of Physical Chemistry|volume=91 | number =15 | pages =4121–4122 | doi=10.1021/j100299a035}}
• {{cite journal | first1=K. D. | last1=Gibson |first2=Harold A. | last2=Scheraga |title=Exact calculation of the volume and surface area of fused hard-sphere molecules with unequal atomic radii |year=1987 | journal= Molecular Physics|volume=62 | number=5 | pages=1247–1265 | bibcode=1987MolPh..62.1247G |doi=10.1080/00268978700102951}}
• {{ cite journal | first1=Michel | last1=Petitjean |title=On the analytical calculation of van der Waals surfaces and volumes: some numerical aspects |journal= Journal of Computational Chemistry|year=1994 | volume=15 | number=5 | pages=507–523 | doi=10.1002/jcc.540150504}}
• {{cite journal | first1=J. A. | last1=Grant |first2=B. T. | last2=Pickup |title=A Gaussian description of molecular shape |journal= The Journal of Physical Chemistry|year=1995 | volume=99 | number= 11 |doi=10.1021/j100011a016 |pages=3503–3510}}
• {{cite journal | first1= Jan | last1=Busa | first2=Jozef | last2=Dzurina |first3=Edik | last3=Hayryan | first4=Shura | last4=Hayryan |title=ARVO: A fortran package for computing the solvent accessible surface area and the excluded volume of overlapping spheres via analytic equations |journal= Computer Physics Communications|bibcode=2005CoPhC.165...59B |year=2005 | volume=165 | issue=1 | pages=59–96 | doi=10.1016/j.cpc.2004.08.002}}

{{Commons category|Spherical caps}}

• {{MathWorld |id=SphericalCap |title=Spherical cap}} Derivation and some additional formulas.
• [http://formularium.org/?go=81 Online calculator for spherical cap volume and area].
• [http://mathforum.org/dr.math/faq/formulas/faq.sphere.html#spherecap Summary of spherical formulas].

Category:Spherical geometry