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 = 2pi R^2(1-sinvarphi)

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 = 2pi R^2(1-costheta)

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^2theta}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^2theta}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}}{Gammaleft(frac{n}{2} + 1right)} 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.

The post Area and volume of hypersphere cap first appeared on John D. Cook.

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