Answer to Question #39983 in Math for Iman

Question

Let A and B be two great circles in a sphere S^2={x^2+y^2+z^2=1} such that A intersects B in two and only two points (recall that any two great circles intersect exactly twise). Let X be a finite set of points in A such that the arc lenght between each two of these points is equal and the cardinality of X is prime, i.e X contains prime number of elements.. For example if X contains 5 points, then the arc between each two of the 5 points subtending an angle 72 degree. Similarly, let Y be a finite set of points in B with equal arc length between each two points. Suppose also #B is prime and distinct from #A.

The question is: Can we define a homotopy between X and Y so that X is homotopic to Y. If so, how

Thank you in advance