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Let $p(x)= \pi A_\alpha \pi^{-1}(x) = y$, where $$A_\alpha = \begin{pmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\\ \end{pmatrix}$$ and $\pi:S\to C=\{z=x_1+ix_2\}\cup\infty$.

Show that $y = p(x)$ is a linear fractional transformation. I am a little confused on how to start this problem.

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What is the function $\pi$? –  SL2 Oct 2 '11 at 3:18
Im so sorry I forgot to add that. $\pi : S ----> C$ = { z= x_{1} + i x_2} $\bigcup$ $\infty$ –  Marge Oct 2 '11 at 3:21
Should we assume $\pi$ is also a lft? If so, then all you need is that inverses of and products of lft's are lft's... –  anon Oct 2 '11 at 3:46
It is not stated whether or not $\pi$ is a lft or not. I think that may be the reson I am confused. –  Marge Oct 2 '11 at 3:52
Marge, please tell a bit more. Is $S$, perhaps, the Riemann Sphere and $\pi$ the stereographic projection? This doesn't quite make sense given that $A$ looks like a 2D-rotation, but, pray, tell us what $S$ is? The reason I ask is that it is a standard exercise to show that rotations of the Riemann Sphere correspond to fractional linear transformations of the extended plane. –  Jyrki Lahtonen Oct 2 '11 at 6:19

1 Answer 1

I base my answer on the guess that $\pi$ is the stereographic projection from the Riemann sphere $S$ to the extended complex plane $\mathbf{C}\cup\{\infty\}$. I'm assuming that the equator of the sphere is on the complex plane. Align the 3-axes as follows: let the $x$-axis coincide with the real axis, $y$-axis with the imaginary axis, and $z$-axis stick out of the complex plane. I am further assuming that the exercise is about showing that a rotation about any of these 3 axes corresponds to a LFT of the complex plane. Here's a plan written in terms of extended hints:

Exercise #1: Check that a rotation about the $z$-axes yields an LFT. Well, this rotation by the angle $\phi$, call it $R(\phi)$ amounts to multiplication by $e^{i\phi}$.

Exercise #2: Let $\psi$ be a 90 degree rotation about the $y$-axis. In the 3D-coordinates this is $\psi(x,y,z)=(-z,y,x)$. Verify that the mapping $\pi\circ\psi\circ\pi^{-1}$ is a LFT.

Exercise #3: Show that an arbitrary rotation about the $x$-axis corresponds to a LFT. Hint: The LFTs form a group, right? What can you say about the 3D-mapping $\psi\circ R(\phi)\circ \psi^{-1}$?

Exercise #4: The same as in exercise #2, but this time we rotate about the $x$-axis, so instead of $\psi$ we look at $\tau(x,y,z)=(x,-z,y)$. Alternatively you can use a little bit of geometric thinking and combine ideas from exercises #1 and #3.

Exercise #5: The same as in exercise #4, but use $\tau$ instead of $\psi$ to show that an arbitrary rotation about the $y$-axis corresponds to a LFT.

If the question was only about those '2D' rotations then you are done. If you want to show the same for all the rotations of $S$ about an arbitrary axis, then I give you one more ...

Hint #6: Have you heard of Euler angles?

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I apologize, if this was off the mark. Also I did my utmost to steer away from messy calculations involving the stereographic projection. Therefore I use a bit of geometric thinking (or matrix algebra, or conjugations in the group of rotations) to reduce to a few selected special cases, where the effect of the stereographic projection is hopefully easy to calculate. It really depends on your background, whether this is understandable. –  Jyrki Lahtonen Oct 2 '11 at 8:42
Thanks, @anon. If only we could figure out what the question was. –  Jyrki Lahtonen Oct 2 '11 at 9:47
@Jyrki, thanks for your answer. I do not really understand it at the moment. I am having difficulty understanding your exercise 2 and 3. Do you have any suggestions on how to understand it? –  Marge Oct 2 '11 at 10:56
@Marge: Have you figured out a formula for the projection $\pi$ and its inverse? Perhaps you were given them? Anyway, presumably $\pi^{-1}(u+iv)=(x(u,v),y(u,v),z(u,v))$ for some functions $x,y,z$. In exercise #2 you should calculate $\pi(-z(u,v),y(u,v),x(u,v))=a(u,v)+i b(u,v)$ and then try to find a LFT $p$ such that $p(u+iv)=a(u,v)+i b(u,v)$. –  Jyrki Lahtonen Oct 2 '11 at 11:17
@Marge I haven't done that calculation in ages, so I don't remember the answer. Anyway, the LFT of exercise #2 should map $i\mapsto i$, $1\mapsto \infty$ and $\infty\mapsto-1$, because the sphere is rotating about the imaginary axis. Check whether that LFT matches with the formula for $a(u,v)$ and $b(u,v)$ that you get. –  Jyrki Lahtonen Oct 2 '11 at 18:05

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