"Averaging on a quotient of a sphere"

Lyle Ramshaw

Note #2002-003, February 28, 2002. 





In work at SRC on calibrating cameras automatically, the problem arose of computing "averages" or "centers of gravity" in the quotient space S^3/H, where S^3 denotes the 3-sphere (sitting in 4-space) and H is a certain subgroup of S^3 of size 48.  So the points of the 3-sphere are being identified into cosets, each of size 48.  This note considers two easier related problems: computing averages either



      - in S^n, the n-sphere itself, with no identification

        going on, or



      - in the space S^n/{+1,-1}, the n-sphere in which each 

        point is identified with its antipode, forming a coset 

        of size 2.



These easier problems have fairly satisfactory solutions. But those solutions don't seem to extend to handle cosets of size greater than 2.