$n$-sphere

Let ${\mathrm{S}}_n(r)$ be an $n$-dimensional hypersphere, or $n$-sphere, of radius $r$ centered at the origin:

$$\begin{array}{r}{\mathrm{S}}_n(r)=\text{setb}\vec{x}\in {\mathbb{R}}^n\text{norm}{\vec{x}}_2=r\end{array}$$

The volume and surface area of are given by formulas involving the Gamma function $\Gamma \text{p}\text{dummyarg}$:

$$\begin{array}{rlrlrl}{\mathrm{V}}_n\text{p}r& =\frac{{\pi }^{\frac{n}{2}}}{\mathrm{\Gamma }\text{p}\frac{n}{2}+1}{r}^n& & & {\mathrm{A}}_n\text{p}r& =\frac{2{\pi }^{\frac{n}{2}}}{\mathrm{\Gamma }\text{p}\frac{n}{2}}{r}^{n-1}\end{array}$$

When the radius is left out it is implied to be one, so $S_n$, $V_n$ and $A_n$ are the unit $n$-sphere and its volume and surface area respectively.

Distance Metrics

There are two commonly used distance functions on the unit $n$-sphere, the chord distance, $d_{\mathrm{chord}}$, and arc length $d_{\mathrm{arc}}$.

The chord length is simply the metric inherited from the surrounding ${\mathbb{R}}^n$ space

$$\begin{array}{r}{d}_{\mathrm{chord}}(\vec{a},\vec{b})=\text{norm}{\vec{a}-\vec{b}}_2\end{array}$$

The arc length, $d_{\mathrm{arc}}$, or great-circle distance, is the length of the shortest path between two points on ${\mathrm{S}}^n$. In one dimension ${\mathrm{S}}^1=\{-1,1\}$ and $d_{\mathrm{arc}}$ is ill-defined because there is no connecting path on ${\mathrm{S}}^1$. A natural extension in this case is to define $d_{\mathrm{arc}}\in \{0,\pi \}$. We can compute $d_{\mathrm{arc}}$ as

$$\begin{array}{r}{d}_{\mathrm{arc}}\text{p}\vec{a},\vec{b}={\cos }^{-1}\text{p}\vec{a}\cdot \vec{b}\end{array}$$

While not technically a distance function, another popular function is the 'cosine distance':

$$\begin{array}{r}{d}_{\cos }(\vec{a},\vec{b})=1-\vec{a}\cdot \vec{b}\end{array}$$

We can geometrically interpret the various distances by considering the plane spanned by $\vec{a}$ and $\vec{b}$ intersecting ${\mathrm{S}}_n$:

The distances $d_{\mathrm{arc}}$ and $d_{\mathrm{chord}}$ are related by

$$\begin{array}{rlrlrl}{d}_{\mathrm{chord}}& =2\sin \text{p}\frac{d_{\mathrm{arc}}}{2}& & & d_{\mathrm{arc}}& =2{\sin }^{-1}\text{p}\frac{d_{\mathrm{chord}}}{2}\end{array}$$

for small distances they are good approximations of each other with an error of $O(d^3)$.

$d_{\mathrm{arc}}$ Distribution

Consider the uniform distribution on the $n$-sphere. Uniform here taken to mean the natural Lebesgue measure. An elegant procedure to sample from this distribution is by generating standard normal random vectors and normalizing them:

def sample_sphere(count=1, n=1, r=1, rng=np.random.default_rng()):
    x = rng.standard_normal((count, n))
    return r * x / np.linalg.norm(x, axis=1)[:, np.newaxis]

Draw a pair of vectors $\vec{a}$, $\vec{b}$ from the unit $n$-sphere and consider their distance $d_{\mathrm{arc}}\text{p}\vec{a},\vec{b}$. Let's do this numerically by generating many distances and plotting the histogram. We do this for a number of dimensions to see how the distribution evolves:

We see the exceptional bimodal behaviour in one dimension, in two dimensions the distances are uniformly distributed, and with higher dimensions the distribution converges to $\frac{\pi }{2}$. The value $\frac{\pi }{2}$ is the distance from a pole to the equator. Intuitively as we increase the number of dimensions there will be more space orthogonal to a given vector.

My first guess was a Beta distribution with parameters $\alpha =\beta =n-1$. This matches the behaviour at dimensions $1$ and $2$, and behaves similarly for higher dimensions. Unfortunately, for $n=3$ and higher it is very subtly wrong, as can be seen when we overlay it on a high resolution histogram

So let's find the true distribution. Given a point $\vec{a}\in {\mathrm{S}}^n$, the set of points a distance $d$ away from $\vec{a}$ is a $n-1$-sphere of radius $\sin d$, that is ${\mathrm{S}}^{n-1}\text{p}\sin d$. The infinitesimal probability of hitting this set is (see appendix for derivation)

$$\begin{array}{rl}{f}_n(d)& =\frac{{\mathrm{A}}_{n-1}\text{p}\sin \theta }{{\mathrm{A}}_n\text{p}1}=\frac{{\sin }^{n-2}d}{B\text{p}\frac{1}{2},\frac{n}{2}-\frac{1}{2}}\end{array}$$

where $B$ is the Beta function

$$\begin{array}{r}B\text{p}a,b=\frac{\Gamma \text{p}a\Gamma \text{p}b}{\Gamma \text{p}a+b}\end{array}$$

For consecutive values of $n$ the probability density function $f_n\text{p}{d}_{\mathrm{arc}}$ looks like

References

• S. Li (2011). Concise Formulas for the Area and Volume of a Hyperspherical Cap. https://dx.doi.org/10.3923/ajms.2011.66.70
• Panagiotis Sidiropoulos (2014). N-sphere chord length distribution. https://arxiv.org/abs/1411.5639v1
• J.C. Baez, G. Egan, J.D. Cook, D. Piponi (2018). Random Points on a Sphere (Part 1) https://johncarlosbaez.wordpress.com/2018/07/10/random-points-on-a-sphere-part-1/

Appendix: Derivation

$$\begin{array}{rl}{f}_n(d)& =\frac{{\mathrm{A}}_{n-1}\text{p}\sin \theta }{{\mathrm{A}}_n\text{p}1}=\frac{\text{p}\frac{2{\pi }^{\frac{n-1}{2}}}{\mathrm{\Gamma }\text{p}\frac{n-1}{2}}\text{p}{\sin \theta }^{n-2}}{\text{p}\frac{2{\pi }^{\frac{n}{2}}}{\mathrm{\Gamma }\text{p}\frac{n}{2}}{1}^{n-1}}\\ & =\frac{1}{\sqrt{\pi }}\frac{\mathrm{\Gamma }\text{p}\frac{n}{2}}{\mathrm{\Gamma }\text{p}\frac{n-1}{2}}{\sin }^{n-2}\theta =\frac{\mathrm{\Gamma }\text{p}\frac{n}{2}}{\mathrm{\Gamma }\text{p}\frac{1}{2}\mathrm{\Gamma }\text{p}\frac{n}{2}-\frac{1}{2}}{\sin }^{n-2}d\\ & =\frac{{\sin }^{n-2}d}{B\text{p}\frac{1}{2},\frac{n}{2}-\frac{1}{2}}\end{array}$$

https://math.stackexchange.com/questions/1246748/distribution-of-an-angle-between-a-random-and-fixed-unit-length-n-vectors