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.

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. 
