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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,

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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 $Q$.

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Thanks Christian, but to use your more precise language, if $P,Q,T$ are points on $S^2$ and $P$ is the starting point, $Q$ is some point whose distance on the surface of the sphere is closer to $P$ then $T$, which is also on the sphere some other distance away. Then are the cardinalities for the sets of mappings between $P$ and $Q$, and for $P$ and $T$ different. I know there are an infinity of mappings for either set, but I was wondering if one had a larger infinity of them then the other. Thanks again. –  kηives Jun 25 '12 at 15:46
    
@kηives: The cardinalities are the same: It's the number of points in the interval $(0,1)$ or in all of ${\mathbb R}$. –  Christian Blatter Jun 25 '12 at 15:52
    
Can you edit that into your answer and I'll check it off ;) –  kηives Jun 25 '12 at 16:09

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