I am wondering about the number of mappings from a point on a sphere to a neighboring point, and a not so neighboring point.
If I take a 2-sphere, and place it on some $x,y,z$-axis and fix those so that the center is at the origin, then I draw a point on the surface close, but some finite distance, from one of the axis, say the $x$, are there distinct cardinalities associated with the sets of change of basis mappings from that point to the point on the sphere on the $x$-axis then there are mappings to, say, the point on the sphere on the $y$-axis? Another way, are their more rotations associated to either set of rotations from that point to putting it on the $x$-axis then on the $y$ just because of its relative location?
Thanks in advance,