Editing Spherical cap

{{short description|Section of a sphere}}
[[File:Spherical cap diagram.tiff|thumb|An example of a spherical cap in blue (and another in red)]]

In [[geometry]], a '''spherical cap''' or '''spherical dome''' is a portion of a [[sphere]] or of a [[ball (mathematics)|ball]] cut off by a [[plane (mathematics)|plane]]. It is also a [[spherical segment]] of one base, i.e., bounded by a single plane. If the plane passes through the [[center (geometry)|center]] of the sphere (forming a [[great circle]]), so that the height of the cap is equal to the [[radius]] of the sphere, the spherical cap is called a ''[[Sphere#Hemisphere|hemisphere]]''.

==Volume and surface area==
The [[volume]] of the spherical cap and the area of the curved surface may be calculated using combinations of
* The radius <math>r</math> of the sphere
* The radius <math>a</math> of the base of the cap
* The height <math>h</math> of the cap
* The [[Spherical coordinate system|polar angle]] <math>\theta</math> between the rays from the center of the sphere to the apex of the cap (the pole) and the edge of the [[disk (mathematics)|disk]] forming the base of the cap.

These variables are inter-related through the formulas
<math> a = r \sin \theta</math>, <math>h = r ( 1 - \cos \theta )</math>, <math>2hr = a^2 + h^2</math>,
and <math>2 h a = (a^2 + h^2)\sin \theta</math>.

{| class="wikitable"
!
! Using <math>r</math> and <math>h</math>
! Using <math>a</math> and <math>h</math>
! Using <math>r</math> and <math>\theta</math>
|-
! Volume
| <math>V = \frac {\pi h^2}{3} (3r-h)</math> <ref name="handbook">{{citation|title=Handbook of Mathematics for Engineers and Scientists|first1=Andrei D|last1=Polyanin|first2=Alexander V.|last2=Manzhirov|publisher=CRC Press|year=2006|isbn=9781584885023|page=69|url=https://books.google.com/books?id=ge6nk9W0BCcC&pg=PA69}}.</ref>
| <math>V = \frac{1}{6}\pi h (3a^2 + h^2)</math>
| <math>V = \frac{\pi}{3} r^3 (2+\cos\theta) (1-\cos\theta)^2 </math>
|-
! Area
| <math>A = 2 \pi r h</math><ref name="handbook"/>
| <math>A =\pi (a^2 + h^2)</math>
| <math>A=2 \pi r^2 (1-\cos \theta)</math>
|-
! Constraints
| <math> 0 \leq h \leq 2 r </math>
| <math> 0 \leq a, \; 0 \leq h </math>
| <math> 0 \leq \theta \leq \pi, \; 0 \leq r</math>
|}
If <math>\phi</math> denotes the [[latitude]] in [[geographic coordinates]], then <math>\theta+\phi = \pi/2 = 90^\circ\,</math>, and <math>\cos \theta = \sin \phi</math>.

=== Deriving the surface area intuitively from the spherical sector volume ===

Note that aside from the calculus based argument below, the area of the spherical cap may be derived from [[Spherical sector#Volume|the volume <math>V_{sec}</math> of the spherical sector]], by an intuitive argument,<ref>{{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}}</ref> as
:<math>A = \frac{3}{r}V_{sec} = \frac{3}{r} \frac{2\pi r^2h}{3} = 2\pi rh\,.</math>
The intuitive argument is based upon summing the total sector volume from that of infinitesimal [[Tetrahedron|triangular pyramids]]. Utilizing the [[Pyramid (geometry)#Volume|pyramid (or cone) volume]] formula of <math>V = \frac{1}{3} bh'</math>, where <math>b</math> is the infinitesimal [[area]] of each pyramidal base (located on the surface of the sphere) and <math>h'</math> is the height of each pyramid from its base to its apex (at the center of the sphere). Since each <math>h'</math>, in the limit, is constant and equivalent to the radius <math>r</math> of the sphere, the sum of the infinitesimal pyramidal bases would equal the area of the spherical sector, and: 
:<math>V_{sec} = \sum{V} = \sum\frac{1}{3} bh' = \sum\frac{1}{3} br = \frac{r}{3} \sum b = \frac{r}{3} A</math>

===Deriving the volume and surface area using calculus ===
[[File:Spherical cap from rotation.svg|thumb|Rotating the green area creates a spherical cap with height <math>h</math> and sphere radius <math>r</math>.]]
The volume and area formulas may be derived by examining the rotation of the function 
:<math>f(x)=\sqrt{r^2-(x-r)^2}=\sqrt{2rx-x^2}</math> 
for <math>x \in [0,h]</math>, using the formulas the [[Surface of revolution|surface of the rotation]] for the area and the [[Solid of revolution|solid of the revolution]] for the volume.
The area is
:<math>A = 2\pi\int_0^h f(x) \sqrt{1+f'(x)^2} \,dx </math>
The derivative of <math>f</math> is
:<math>f'(x) = \frac{r-x}{\sqrt{2rx-x^2}} </math>
and hence
:<math>1+f'(x)^2 = \frac{r^2}{2rx-x^2} </math>
The formula for the area is therefore
:<math>A = 2\pi\int_0^h \sqrt{2rx-x^2} \sqrt{\frac{r^2}{2rx-x^2}} \,dx
         = 2\pi \int_0^h r\,dx
         = 2\pi r \left[x\right]_0^h
         = 2 \pi r h </math>
The volume is
:<math>V = \pi \int_0^h f(x)^2 \,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)</math>

== Moment of inertia ==
The moments of inertia of a spherical cap (where the z-axis is the symmetrical axis) about the principal axes (center) of the sphere are:

:<math>J_{zz,\text{cap}} = \frac{m h \left( 3 h^2 - 15 h R + 20 R^2 \right)}{10 \left( 3 R - h \right)} 
</math>

:<math>J_{xx,\text{cap}} = J_{yy, \text{cap}} =\frac{m \left( -9 h^3 + 45 h^2 R - 80 h R^2 + 60 R^3 \right)}{20 \left( 3 R - h \right)} </math>

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

==Applications==
===Volumes of union and intersection of two intersecting spheres===

The volume of the [[union (set theory)|union]] of two intersecting spheres
of radii <math>r_1</math> and <math>r_2</math> is
<ref>{{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}}</ref>

:<math> V = V^{(1)}-V^{(2)}\,,</math>

where

:<math>V^{(1)} = \frac{4\pi}{3}r_1^3 +\frac{4\pi}{3}r_2^3</math>

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

:<math>V^{(2)} = \frac{\pi h_1^2}{3}(3r_1-h_1)+\frac{\pi h_2^2}{3}(3r_2-h_2)</math>

the sum of the volumes of the two spherical caps forming their intersection. If <math>d \le r_1+r_2</math> is the
distance between the two sphere centers, elimination of the variables <math>h_1</math> and <math>h_2</math> leads
to<ref>{{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}}</ref><ref>{{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}}</ref>

:<math>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)\,.</math>

=== Volume of a spherical cap with a curved base ===

The volume of a spherical cap with a curved base can be calculated by considering two spheres with radii <math>r_1</math> and <math>r_2</math>, separated by some distance <math>d</math>, and for which their surfaces intersect at <math>x=h</math>. That is, the curvature of the base comes from sphere 2. The volume is thus the difference between sphere 2's cap (with height <math>(r_2-r_1)-(d-h)</math>) and sphere 1's cap (with height <math>h</math>),

<math>\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}
</math>

This formula is valid only for configurations that satisfy <math>0<d<r_2</math> and <math>d-(r_2-r_1)<h\leq r_1</math>. If sphere 2 is very large such that <math>r_2\gg r_1</math>, hence <math>d \gg h</math> and <math>r_2\approx d</math>, which is the case for a spherical cap with a base that has a negligible curvature, the above equation is equal to the volume of a spherical cap with a flat base, as expected.

=== Areas of intersecting spheres ===

Consider two intersecting spheres of radii <math>r_1</math> and <math>r_2</math>, with their centers separated by distance <math>d</math>. They intersect if

:<math>|r_1-r_2|\leq d \leq r_1+r_2</math>

From the law of cosines, the polar angle of the spherical cap on the sphere of radius <math>r_1</math> is

:<math>\cos \theta = \frac{r_1^2-r_2^2+d^2}{2r_1d}</math>

Using this, the surface area of the spherical cap on the sphere of radius <math>r_1</math> is

:<math>A_1 = 2\pi r_1^2 \left( 1+\frac{r_2^2-r_1^2-d^2}{2 r_1 d} \right)</math>

=== 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 <math>r</math>, and caps with heights <math>h_1</math> and <math>h_2</math>, the area is

:<math>A=2 \pi r |h_1 - h_2|\,,</math>

or, using geographic coordinates with latitudes <math>\phi_1</math> and <math>\phi_2</math>,<ref>{{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}}</ref>

:<math>A=2 \pi r^2 |\sin \phi_1 - \sin \phi_2|\,,</math>

For example, assuming the Earth is a sphere of radius 6371&nbsp;km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016<ref>{{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}}</ref>) is {{math|1= 2''π''{{thinsp}}&sdot;{{thinsp}}6371<sup>2</sup>{{thinsp}}{{abs|sin 90° &minus; sin 66.56°}}}} = {{convert|21.04|e6km2|e6mi2|abbr=unit}}, or {{math|1= 0.5{{thinsp}}&sdot;{{thinsp}}{{abs|sin 90° &minus; 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 [[circular symmetry|circularly symmetric]] (having an axis of rotation), and likewise the [[ellipsoidal dome]] is derived from the [[ellipsoid]].

=== Hyperspherical cap ===

Generally, the <math>n</math>-dimensional volume of a hyperspherical cap of height <math>h</math> and radius <math>r</math> in <math>n</math>-dimensional Euclidean space is given by:<ref name="S-Li">{{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}}</ref>
<math display="block">V = \frac{\pi ^ {\frac{n-1}{2}}\, r^{n}}{\,\Gamma \left ( \frac{n+1}{2} \right )} \int_{0}^{\arccos\left(\frac{r-h}{r}\right)}\sin^n (\theta) \,\mathrm{d}\theta</math>
where <math>\Gamma</math> (the [[gamma function]]) is given by <math> \Gamma(z) = \int_0^\infty t^{z-1} \mathrm{e}^{-t}\,\mathrm{d}t </math>.

The formula for <math>V</math> can be expressed in terms of the volume of the unit [[n-ball]] <math display="inline">C_n =  \pi^{n/2} / \Gamma[1+\frac{n}{2}]</math> and the [[hypergeometric function]] <math>{}_{2}F_{1}</math> or the [[regularized incomplete beta function]] <math>I_x(a,b)</math> as
<math display="block">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),</math>

and the area formula <math>A</math> can be expressed in terms of the area of the unit n-ball <math display="inline">A_{n}={2\pi^{n/2}/\Gamma[\frac{n}{2}]}</math> as
<math display="block">A =\frac{1}{2}A_{n} \, r^{n-1} I_{(2rh-h^2)/r^2} \left(\frac{n-1}{2}, \frac{1}{2} \right),</math>
where <math>0\le h\le r </math>.

A. Chudnov<ref name="Minimax-86">{{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}}</ref> derived the following formulas:
<math display="block"> A = A_n r^{n-1} p_ { n-2 } (q),\, V = C_n r^{n} p_n (q) , </math> where 
<math display="block"> q = 1-h/r (0 \le q \le 1 ), p_n (q) =(1-G_n(q)/G_n(1))/2 , </math>
<math display="block"> G _n(q)= \int _0^q (1-t^2) ^ { (n-1) /2 } dt .</math>

For odd <math> n=2k+1 </math>:
<math display="block"> G_n(q) = \sum_{i=0}^k (-1) ^i \binom k i \frac {q^{2i+1}} {2i+1} .</math>

==== Asymptotics ====

If <math> n \to \infty </math> and <math>q\sqrt n = \text{const.}</math>, then <math> p_n (q) \to 1- F({q \sqrt n}) </math> where <math> F() </math> is the integral of the [[standard normal distribution]].<ref name= "Game-91">{{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}}</ref>

A more quantitative bound is <math> A/(A_n r^{n-1}) = n^{\Theta(1)} \cdot [(2-h/r)h/r]^{n/2} </math>.
For large caps (that is when <math>(1-h/r)^4\cdot n = O(1)</math> as <math>n\to \infty</math>), the bound simplifies to <math>n^{\Theta(1)} \cdot e^{-(1-h/r)^2n/2} </math>.<ref>{{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 }}</ref>

== See also ==
{{Portal|Maths}}
* [[Circular segment]] — the analogous 2D object
* [[Solid angle]] — contains formula for n-sphere caps
* [[Spherical segment]]
* [[Spherical sector]]
* [[Spherical wedge]]

== References ==
{{reflist}}

== Further reading ==
* {{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}}

== External links ==
{{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]]