Intersection of two spherical caps in $(n+1)$-dimensional Euclidian space

Question

I want to compute the intersection area of two spherical caps whose tops are placed on two orthogonal axes in $\mathbb{R}^{n+1}$. In spherical coordinates, the surface element is given by

$$ \,dS = \sin^{n-1}(\theta_1) \sin^{n-2}(\theta_2)\cdots \sin(\theta_{n-1})\,d\theta_{1}\,d\theta_{2}\cdots \,d\theta_{n-1}\,d\theta_{n},$$ with $\theta_{n} \in [0,2\pi)$ and the other angles in $[0, \pi]$.

My question is: what should be the integration limits in

$$\text{Area} = \idotsint \limits_{?} \,dS$$ to computed this intersection surface?

Thanks in advance!

Answer 1

You haven't written whether you want to consider two hyperspherical caps of the same height.

The answer to your question, even with different heights, is given as the intersection of two hyperspherical caps in $n$ dimensions (sorry for changing that, you wanted $n+1$) which can be found in the 2014 article by Lee and Kim, http://ie.kaist.ac.kr/...

Since you are considering rectangular axes of the caps, the cases 6,7, and 8 in this article apply. If also the heights of your two caps equal, we have case 8, which can be explained as follows:

Half of the intersection area you are looking for is actually the area of one of the hyperspherical caps, cut by a hyperplane passing through the origin and towards the "central" intersection of the two caps. Lets call this area (following Lee and Kim, slightly simplified)

$$ J_n^{\theta} (r) $$

where $n$ are the dimensions, $r$ the radius, and $\theta$ is the colatitude angle of your cap, i.e. $h = (1-\cos \theta) r $ is the cap height. We need $\theta > \pi / 4$ for the caps to intersect at all. Lee and Kim also define $\theta_{min}$ as the angle between the cap axis and the hyperplane which clearly here is $\pi / 4$. Then the surface area can be obtained by integrating the area of the ( n − 1 ) - dimensional hyperspherical cap of radius $r \sin(\Theta)$ with the colatitude angle $\arccos(\cot(\Theta))$ over the arc rd$\Theta$ for $\Theta$ from $\pi/4$ to $\theta$. So the area of the intersection of your two hyperspherical caps (with same heights) is

$$ 2 J_n^{\theta} (r) = 2 \int_{\pi/4}^{\theta} A_{n-1}^{\arccos(\cot(\Theta))}(r \sin\Theta) rd \Theta $$ where the area of an n-dimensional hyperspherical cap with colatitude angle $\theta$ is given by $$ A_{n}^{\theta}(r) = \frac12 A_{n}(r) I_{sin^2 \theta} (\frac{n-1}{2}, \frac12) $$ where $A_{n}(r) $ is the (well known) full area of the $n$-dimensional sphere and $I$ is the regularized incomplete beta function.

Following your question on the integration boundaries:

Since $A_{n}(r) = A_{n}(1) r^{n-1}$, we have to deal with $A_{n-1}(r \sin\Theta) = A_{n-1}(1) r^{n-2} (\sin\Theta)^{n-2}$ and hence the above integration reduces to one (!) intricate integration over $\Theta$ and otherwise to (n-2) unrestricted integrations over all the other angles, which lead to the area of the unit sphere, $A_{n-1}(1) $.