0
$\begingroup$

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,

1 Answers 1

0

If I understand correctly there is no "change of basis" occurring here at all, but you are asking about rotations moving some given point $P\in S^2$ to some other given point $Q\in S^2$, where $Q$ happens to lie on one of the coordinate axes.

Now in any case there is an infinity of such rotations: If $P\ne Q$ then any plane $\pi$ through $P$ and $Q$ intersects $S^2$ in a circle. Let $M_\pi$ be the center of this circle. There is a rotation with axis $a:=O\vee M_\pi$ mapping $P$ onto $Q$.

  • 0
    Can you edit that into your answer and I'll check it off ;)2012-06-25