A unit n-sphere is defined as $$\mathcal{S}^n = \{\mathbf{p} \in \mathbb{R}^{n+1}: \|\mathbf{p}\| = 1\}$$

The distance between two points $\mathbf{p}$, $\mathbf{q}$ on $\mathcal{S}^n$ is the great-circle distance: $$\rho(\mathbf{p},\mathbf{q}) = \arccos(\mathbf{p}^T \mathbf{q})$$ where $\arccos(\cdot): [-1,1] \to [0,\pi]$ is the inverse cosine function.

Given a set of points on $\mathcal{S}^n$, I would like to compute the mean (centroid) of these points. Since the surface of a unit n-sphere is not a Euclidean space, I guess we cannot use arithmetic mean in this case.

Question: how do we compute this mean point $\boldsymbol{\mu} \in \mathcal{S}^n$, taking into account the spherical geometry?

1more comment