Editing Spherical cap (section)

== 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>