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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.

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Not a complete answer, but a pointer

Sam Buss wrote an article on this topic (at least for the 2-sphere...I don't recall all the details) for the ACM Transactions on Graphics about 10-15 years ago.

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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).

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