Talk:Spherical cap

Proof of area and volume

Archimedes made geometric proofs of the volume and area of the spherical cap. The formulas can also be shown using calculus (as seen in the article). Would a section on Archimedes' proofs and a section on the calculus proofs be worthwhile to have on the page? — Preceding unsigned comment added by 212.93.55.116 (talk) 14:32, 27 November 2018 (UTC)

I think this would be a good idea since a simple intuitive non-calculus proof is possible. If we use the same method that Archimedes used to find the surface area of a sphere (a sum of thin strips), we can calculate the surface area of the spherical cap directly. Then we can reason to the volume using the method given on the main page by subtracting the volume of a cone.Wrose31 (talk) 10:11, 8 June 2021 (UTC)

$0<h<r$

$V_{cone}=\int _{0}^{r-h}V_{ball}\left(x\tan(\theta ),n-1\right)dx=\int _{0}^{r-h}V_{ball}\left({\frac {ax}{r-h}},n-1\right)dx={\frac {r-h}{n}}V_{ball}\left(a,n-1\right)$

$V_{sector}={\frac {h}{2r}}V_{ball}$ (only at n = 3)

$V_{dome}=V_{sector}-V_{cone}=V_{sector}\left(r,n\right)-{\frac {r-h}{n}}V_{ball}\left(a,n-1\right)$

Have you any simple ideas for $V_{sector}\left(r,n\right)$? MashiraV (talk) 00:50, 1 May 2023 (UTC)

Proposed merger

Spherical cap and Dome (mathematics) are the same mathematical object. The two pages should be merged. Unfortunately they use different notation. Spherical cap seems currently to be the better article and probably has the better name. -Ian Jackson 18 Oct 2011

I absolutely agree with Mr. Jackson. The Spherical Cap discussion has a diagram (!) which makes it easier to relate the cap to the sphere. Joe Smart 16 Dec 2011 joe.smart737@gmail.com — Preceding unsigned comment added by 203.87.162.5 (talk) 02:27, 16 December 2011 (UTC)

I have concerns about this suggestion. It depends on the indexing. I approached the problem looking for "spherical cap", and found what I was looking for. However, had I been looking for information on the volume of a dome, I would not have thought of using the term "spherical cap" (nor vice versa). The equations are different in approach, too. If the two can be combined without truncation (and the result would be some duplication), and if both terms can be found in a search, then combine. But simpler to leave as is.

Resolved

Colors!! Not visible to people with typical 'color blindness' issues.

The illustration of the sectioned sphere on the page: http://en.wikipedia.org/wiki/Spherical_cap shows the sphere divided into two pieces. The illustration has the caption: "The spherical cap is the purple section." The text also states: "The blue section of the illustration is also a spherical cap."

This is fine for people with 'normal' color vision, but for the large percentage of people with the typical vision aberration known as 'red/green' color blindness there is NO DIFFERENCE in the colors. Both sections of the sphere look identically 'blue'.

I recommend you use blue as the color for the cap and use red for the rest of the sphere.

I also recommend that writers and editors of Wikipedia articles become familiar with the guidelines for creating webpages that are 'color equal'. You can 'google' the topic. Here is one link as a starting point: http://99designs.com/designer-blog/2013/04/17/designers-need-to-understand-color-blindness/ — Preceding unsigned comment added by 67.141.130.227 (talk) 21:16, 26 May 2015 (UTC)

Completely agree! I'm working on a fix for this and will try to post it soon it Jhmadden (talk) 23:17, 4 November 2015 (UTC)

hope the new image is better! Jhmadden (talk) 23:37, 4 November 2015 (UTC)

Resolved

Formula based on angle

It may be useful for readers to have the formula for area based on angle between the edge and top of the cap to the centre of the sphere. This is useful when calculating radiation patterns. I think this may be

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

Where θ is the angle between the top of the cap and the edge from the middle of the sphere.Nick Hill (talk) 17:29, 18 September 2015 (UTC)

yes this is a very useful formula for this object. I'm working on a new figure that solves the colorblindness problem and will at a $\theta$ where it needs to be. Jhmadden (talk) 23:19, 4 November 2015 (UTC)

added theta and the additional surface area equation. feel free to elaborate on its derivation if you feel that's necessary too! Jhmadden (talk) 23:39, 4 November 2015 (UTC)

Resolved

Formula for surface area of hyperspherical cap

Where does the formula for the surface area of the hyperspherical cap come from? In particular the index of the incomplete beta function? I can not find it in

References

Answer:Look at the very first figure of the Spherical cap page. Replace there $\theta$ by $\phi$ and note that then $a=r\sin \phi .$ Then compare with the paper of S. Li you cite. You find by easy geometry $\sin ^{2}\phi ={\frac {2hr-h^{2}}{r^{2}}}$ Alexander Kovacec.

Fundamental error in terminology

The subsection Hyperspherical cap of the section Generalizations contains a sentence beginning as follows:

"Generally, the n-dimensional volume of a hyperspherical cap of height h and radius r in n-dimensional Euclidean space is given by ..."

No. A "hyperspherical cap" of a sphere in n-dimensional Euclidean space is the portion of the (n-1)-dimensional hypersurface given by the part of the (n-1)-dimensional sphere lying in, and on one side of, some fixed hyperplane.

Instead the article is discussing the portion of the n-dimensional ball lying on one side of a hyperplane. Perhaps the correct name for this is "segment of a n-dimensional disk" or "segment of an n-dimensional ball". But not "spherical cap".50.205.142.50 (talk) 21:14, 18 June 2020 (UTC)

Using Beta function is conditional

It works only at $h\leqslant r$ , this can be checked by program:

(*Wolfram Language "Mathematica"*)
Clear[a, r, h, n, x, Vsph, Vspi, Vsum, Vseh, Vseb, Vtet, Vint, Fsgi];

Vsph[r_, n_] := r^n \[Pi]^(n/2)/Gamma[n/2 + 1];
Vspi[r_, n_] := Module[{k, p, i, V}, p = Mod[n, 2]; k = (n - p)/2; V = If[p == 1, 2 r, 1];
   If[n <= 1, V, i = +p; While[i < n, V *= 2 \[Pi] r^2/(i + 2); i += 2]; V]];
Fsgi[n_, t_] := Module[{k, p, i, F}, p = Mod[n, 2]; k = (n - p)/2; F = If[p == 1, ArcCos[t], 1 - t];
   If[n <= 0, F, i = -p; While[i < n, F = ((i + 2) F - t (1 - t^2)^((i + 2)/2))/(i + 3); i += 2]; F]];
Vsum[r_, t_, n_] := Module[{k, p, i}, p = Mod[n, 2]; k = (n - p)/2; If[p == 1, 
   \[Pi]^k r^(2 k + 1) (2^k/(2 k + 1)!! - 1/k! Sum[ ((-1)^i t^(2 i + 1))/(2 i + 1) Binomial[k, i], {i, 0, k}]), "--"]];

(*params values*)
n = 3; r = 7; h = r 5/3; Print["n = ", n, "\tr = ", r, "\th = ", h];

Print["iterations"];
N[r Vspi[r, n - 1] Fsgi[n - 1, (r - h)/r]]

Print["sum"];
N[Vsum[r, (r - h)/r, n]]

Print["Hypergeometric2F1"];
Vseh = N[Vsph[r, n] (1/2 - (r - h)/r Gamma[n/2 + 1]/(Sqrt[\[Pi]] Gamma[(n + 1)/2])
       Hypergeometric2F1[1/2, (1 - n)/2, 3/2, ((h - r)/r)^2])]

Print["BetaRegularized"];
Vseb = N[Vsph[r, n]/2 BetaRegularized[(h (2 r - h))/r^2, (n + 1)/2, 1/2]]

Print["Integrate by angle"];
Vtet = N[r Vsph[r, n - 1] Integrate[Sin[x]^n, {x, 0, ArcCos[(r - h)/r]}]]

Print["Integrate by height"];
Vint = N[Integrate[Vsph[Sqrt[r^2 - x^2], n - 1], {x, r - h, r}]]

Print["Given dimensions"];
If[n == 3, a = Sqrt[h (2 r - h)];
  (*height and spherical radius*)
  Print[N[1/3 h^2 \[Pi] (3 r - h)]];
  (*height and base radius*)
  Print[N[(\[Pi] h)/6 (3 a^2 + h^2)]]];

If[n == 2, a = Sqrt[h (2 r - h)];
  (*height with base and spherical radii*)
  Print[N[r^2 ArcCos[(r - h)/r] - a (r - h)]]];

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Program contains one more way for general case, for odd dimensions result without special functions and finite sum, useful in programing with small math libraries:

- finite iteration, at $\textstyle k={\tfrac {n-mod(n,2)}{2}}$ steps:

restriction	steps	expression	parameters
$n\geqslant 1$	final	$V_{dome}(r,h,n)=rV_{ball}(r,n-1)F(n-1,t)$	$p=mod(n-1,2)$, $t=\cos(\theta )={\tfrac {r-h}{r}}$
$i\geqslant -1$	$k$	$F(i+2,t)={\tfrac {1}{i+3}}\left((i+2)F(i,t)-t\left(1-t^{2}\right)^{\frac {i+2}{2}}\right)$	$F(-p,t)={\begin{cases}1-t&p=0\\\arccos(t)&p=1\end{cases}}$
$i\geqslant 0$	$k-p$	$V_{ball}(r,i+2)={\tfrac {2\pi r^{2}}{i+2}}V_{ball}(r,i)$	$V_{ball}(r,p)={\begin{cases}1&p=0\\2r&p=1\end{cases}}$

- finite sum, closed form, odd dimensions:

$V_{dome}(r,h,n)={\begin{cases}--,&p=0\\\pi ^{k}r^{2k+1}\left({\tfrac {2^{k}}{(2k+1)!!}}-{\tfrac {1}{k!}}\sum _{i=k}^{0}{\binom {k}{i}}{\tfrac {(-1)^{i}t^{2i+1}}{2i+1}}\right)&p=1\end{cases}}$

$\textstyle k=(n-p)/2,~p=mod(n,2),~{\binom {k}{i}}={\begin{cases}{\frac {k(k-1)(k-2)\cdot \ldots \cdot (k-i+1)}{i!}}&i\geqslant 1\\1&i=0\end{cases}}$ - binomial coefficient.

The expressions are close to the calculus, the summation to reduce the error starts with small terms, this doubles the number of operations compared to the direct order, and requires additional RAM due to the use of arrays. Or for size optimization, leads the complexity of the calculation to quadratic. For a quick estimate calculation or analytical expressions, it is appropriate to use the direct order $\textstyle \sum _{i=k}^{0}=\sum _{i=0}^{k}$, the argument of the sum is invariant with respect to the substitution of the limit values of the index.

Finite sum way has close form to (1986, USSR Academ. Press, expressions are not implemented in reference) but not conditioned.

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Definition by angle integrating should be extended height integration, to agree with example:

$V_{dome}(r,h,n)=\int \limits _{r-h}^{r}V_{ball}\left(a(x),n-1\right)dx=r\,V_{ball}(r,n-1)\int \limits _{0}^{\arccos \left({\frac {r-h}{r}}\right)}\sin ^{n}\theta \,d\theta$

where $a(x)={\sqrt {r^{2}-x^{2}}}$, $V_{ball}(r,k)={\frac {r^{k}\pi ^{k/2}}{\Gamma \left(k/2+1\right)}}$.

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MashiraV (talk) 16:03, 30 April 2023 (UTC)