Wednesday, September 17, 2014
Distance Between Two Points on an n-sphere
One can show that an arc of a great circle is the shortest distance between two points on an n-sphere. Along the great circle arc through points a and b we choose a point q and an intermediate point p between a and q. We next choose a point p' on a perpendicular great circle offset a small angle δ from the first great circle at p. We then show that the combined length of the two arcs a-p' and p'-q is greater than the arc a-p-q.
We find that the cosine of the angles between two consecutive points are reduced by cos(δ) which corresponds to the angles being increased by small angles μ and ν respectively.
To simplify the expressions for the angles capital letters were used instead of vectors. Note that instead of the n-2 tangent vectors tk one can use an arbitrary tangent vector t in the tangent space for the variation and the results will be unchanged.