Area and volume of hypersphere cap

Posted on by John

A spherical cap is the portion of a sphere above some horizontal plane. For example, the polar ice cap of the earth is the region above some latitude. I mentioned in this post that the area above a latitude φ is

A = 2\pi R^2(1-\sin\varphi)

where R is the earth’s radius. Latitude is the angle up from the equator. If we use the angle θ down from the pole, we get

A = 2\pi R^2(1-\cos\theta)

I recently ran across a generalization of this formula to higher-dimensional spheres in [1]. This paper uses the polar angle θ rather than latitude φ. Throughout this post we assume 0 ≤ θ ≤ π/2.

The paper also includes a formula for the volume of a hypersphere cap which I will include here.

Definitions

Let S be the surface of a ball in n-dimensional space and let An(R) be its surface area.

A_n(R) = \frac{\pi^{n/2}}{\Gamma(n/2)} R^{n-1}

Let Ix(a, b) be the incomplete beta function with parameters a and b evaluated at x. (This notation is arcane but standard.)

I_x(a, b) = \int_0^x t^{a-1}\, (1-t)^{b-1}\, dt

This is, aside from a normalizing constant, the CDF function of a beta(a, b) random variable. To make it into the CDF, divide by B(a, b), the (complete) beta function.

B(a, b) = \int_0^1 t^{a-1}\, (1-t)^{b-1}\, dt

Area equation

Now we can state the equation for the area of a spherical cap of a hypersphere in n dimensions.

A_n^{\text{cap}}(R) = \frac{1}{2}A_n(R)\, I_{\sin^2\theta}\left(\frac{n-1}{2}, \frac{1}{2} \right )

Recall that we assume the polar angle θ satisfies 0 ≤ θ ≤ π/2.

It’s not obvious that this reduces to the equation at the top of the post when n = 3, but it does.

Volume equation

The equation for the volume of the spherical cap is very similar:

V_n^{\text{cap}}(R) = \frac{1}{2}V_n(R)\, I_{\sin^2\theta}\left(\frac{n+1}{2}, \frac{1}{2} \right )

where Vn(R) is the volume of a ball of radius R in n dimensions.

V_n(R) = \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2} + 1\right)} R^n

Related posts

[1] Shengqiao Li. Concise Formulas for the Area and Volume of a Hyperspherical Cap. Asian Journal of Mathematics and Statistics 4 (1): 66–70, 2011.