Area and volume of hypersphere cap
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 is
where R is the earth's radius. Latitude is the angle up from the equator. If we use the angle down from the pole, we get
I recently ran across a generalization of this formula to higher-dimensional spheres in [1]. This paper uses the polar angle rather than latitude . Throughout this post we assume 0 /2.
The paper also includes a formula for the volume of a hypersphere cap which I will include here.
DefinitionsLet S be the surface of a ball in n-dimensional space and let An(R) be its surface area.
Let Ix(a, b) be the incomplete beta function with parameters a and b evaluated at x. (This notation is arcane but standard.)
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.
Now we can state the equation for the area of a spherical cap of a hypersphere in n dimensions.
Recall that we assume the polar angle satisfies 0 /2.
It's not obvious that this reduces to the equation at the top of the post when n = 3, but it does.
Volume equationThe equation for the volume of the spherical cap is very similar:
where Vn(R) is the volume of a ball of radius R in n dimensions.
[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.
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