# Extending "mean of circular quantities" to a sphere of 3 or more dimensions

I am looking for a way to find the mean of a set of unit vectors (or points on a unit sphere), with the same or similar properties to the mean of circular quantities, only extended to 3 (or higher) dimensions. Does this require spherical harmonics or similar, for 3 or higher dimensions?

I am then looking for a way to extend this to quaternions, for the purpose of averaging quaternions without resorting to the SVD, eigendecomposition, or traces and Froebenius norms.

I don’t know if it has all the same properties as the mean of circular quantities, but one quantity with reasonably mean like properties can be found by embedding the $$n$$-sphere in $$\mathbb{R}^n$$, then take the arithmetic mean of the resulting unit vectors, and project back on to the unit $$n$$-sphere.
For example, if you wanted to find the mean of (45 degrees latitude, 0 degrees longitude) and (-45 degrees latitude, 0 degrees longitude), you would convert to rectangular coordinates: (0, $$\frac{\sqrt{2}}{2}$$,$$\frac{\sqrt{2}}{2}$$) and , (0, -$$\frac{\sqrt{2}}{2}$$,$$\frac{\sqrt{2}}{2}$$) then compute the mean: (0,0,$$\frac{\sqrt{2}}{2}$$), project to the unit circle: (0,0,1), and if you wish, revert to spherical coordinates: (0 degrees, 0 degrees).