SRC Technical Note 2002-003

Averaging on a quotient of a sphere

Lyle Ramshaw

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.

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