I want to calculate the surface area of an n-dim hyperspherical cap with radius r=1.
I found S. Li's evidence, but I don't understand his idea:
http://docsdrive.com/pdfs/ansinet/ajms/2011/66-70.pdf
His idea is for $\Phi \leq \pi /2$:
"The area of a hypershperical cap in a n-sphere can be obtained by integrating the surface area of an (n-1)-sphere of radius $\sin (\theta)$ with arc element $d\theta$ over a great circle arc:
$A^{cap}_n=\int_{0}^{\Phi} A_{n-1}(\sin (\theta))\ \,d\theta$
with the area of an n-sphere:
$A_n(r)=\frac{2 \pi^{n/2}}{\Gamma (n/2)} r^{n-1}$
I don't have much knowledge of Ana3 so I would be happy to receive detailed answers.
Thank you very much 🙂
Best Answer
Consider the following image of concentric rings of constant width on a unit sphere:
The area of the circular ring with radius $\theta$ (measured on a great circle arc) and width $\Delta\theta$ can be approximated by $2\pi\sin(\theta)\cdot\Delta\theta$ if $\Delta\theta$ is small. So the area of a cap with radius $\Phi$ (again, measured on a great circle arc) is equal to the integral over the area of all these rings:
$$\int_0^{\Phi} 2\pi\sin(\theta)d\theta.$$
Note that $2\pi\sin(\theta) = A_2(\sin(\theta))$ is just the measure of the circle followed by the ring.
In higher dimensions, the same approach, dividing the area of the $n$-sphere into concentric $(n-1)$-spheres, yields the general formula you found.