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I'm trying to show that a specific Möbius transformation exists, where I have some points that map to some other points. (I don't wanna be too specific here about what goes where, as I don't wanna run the risk of having the insight in solving this myself spoiled).

How does one go about proving the existence of a Möbius transformation? My first idea was to start with the cross-ratio, but I'm very unsure about the theoretical relationship between the cross-ratio and the Möbius transformation. As I understand it, the cross-ratio IS a transformation, so using it will be like assuming that a transformation already exists to prove that a transformation exists. Or is it enough to show that if under a certain mapping, the cross ratio will give a something that fits the definition of a transformation?

Very confused about this, any insights will help!

Thanks in advance!

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    $\begingroup$ The cross-ratio is a Möbius transformation that maps three given (distinct) points to the points $0,1,\infty$. If you want to map three (or fewer) points to pre-assigned images, the cross-ratios give you the existence, and an explicit form. If you want to map more points to pre-assigned images, such a transformation need not exists, but if it does, you obtain it via the cross-ratios too, so you take what the cross-ratios give you, and see whether it does what you want beyond what you put into the cross-ratios. $\endgroup$ – Daniel Fischer Feb 22 '14 at 15:33
  • $\begingroup$ @DanielFischer This would be better suited as an answer. As it says: "Avoid answering questions in comments". $\endgroup$ – Fly by Night Feb 22 '14 at 15:39
  • $\begingroup$ @FlybyNight I'm not sure enough what the question is to decide whether that's an answer. $\endgroup$ – Daniel Fischer Feb 22 '14 at 15:45
  • $\begingroup$ @DanielFischer I would certainly up-vote this if it were an answer. Moreover, it certainly doesn't qualify as a comment because comments should "Ask for more information or suggest improvements". You do neither of these things. What you do do is give a nice answer to the OP's question :o) $\endgroup$ – Fly by Night Feb 22 '14 at 15:48
  • $\begingroup$ @FlybyNight I didn't copy it verbatim to leave you some wiggle-room with respect to upvoting ;) $\endgroup$ – Daniel Fischer Feb 22 '14 at 16:44
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The cross-ratio is a Möbius transformation that maps three given distinct points to $0,1,\infty$, respectively. $\DeclareMathOperator{\CR}{CR}$ (Caution: There are different conventions for the cross-ratio, under one convention $\CR(z,z_1,z_2,z_3)$ maps $z_1$ to $0$ and $z_2$ to $1$, under the other widely used one, it maps $z_1\mapsto 1$ and $z_2\mapsto 0$; there may be yet other conventions, but these would be relatively rarely used, I dare say.)

Thus if you have two triples of distinct points, $(z_1,z_2,z_3)$ and $(w_1,w_2,w_3)$, and want to map $z_i\mapsto w_i$, then

$$T = \CR(\,\cdot\,,w_1,w_2,w_3)^{-1}\circ \CR(\,\cdot\,,z_1,z_2,z_3)$$

is the Möbius transformation that achieves that (and that formula is independent on which convention for the cross-ratio you use). Since a Möbius transformation with three fixed points is the identity, $T$ is the only Möbius transformation that maps $z_i\mapsto w_i$ for $i = 1,2,3$.

If you have fewer than $3$ points to map, there is more than one Möbius transformation mapping $z_i\mapsto w_i$ (assuming the $z_i$ resp. $w_i$ are distinct), you can find one Möbius transformation mapping the given points as desired by introducing arbitrary additional points. If you have more than three points to map, there need not exist a Möbius transformation that achieves the desired mapping. Then you pick triples, construct the Möbius transformation mapping these as desired, and check whether the further demands are met.

Generally,

How does one go about proving the existence of a mobius transformation?

is often answered by explicitly giving the desired transformation. Möbius transformations are relatively simple mappings, it is often easy to explicitly produce the desired transformation(s).

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  • $\begingroup$ Hmmm... This is good... That being said, in my case it's not that easy, as I dont have any given points to map, it's all general. As stated, a Mobius transf. will be uniquely determined by what in does on three points, calculatable by the cross ratio. It it then true that I can map ANY distinct points to ANY distinct points, (3 of them) and ALWAYS end up with a Mobius transf, under the condition that the determinant of the transformation is not 0? cause thats what it seems like to me... $\endgroup$ – JuliusL33t Feb 22 '14 at 18:08
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    $\begingroup$ Yes, you can map any triple of distinct points to another triple of distinct points, and there is, for all pairs of triples, exactly one Möbius transformation doing that. Can you be a bit more specific what your situation is? Do you have to map a disk to a half-plane or something similar? $\endgroup$ – Daniel Fischer Feb 22 '14 at 19:40
  • $\begingroup$ Hi @DanielFischer, given an explicit Mobius transformation, if I have mapped 3 distinct points from a circle in the z-plane to 3 distinct points on the w-plane, can I conclude immediately that the entire pre-image circle maps to the entire image circle? I'm thinking "yes", since 3 distinct, non-collinear points describe a unique circle. I'm having a little bit of trouble going from "there's a unique Mobius tranformation for this action" to "yes, the whole preimage circle maps to the whole image circle. Thanks, $\endgroup$ – User001 Aug 9 '15 at 2:27
  • $\begingroup$ I guess I am trying to say that the image is guaranteed to be a circle that passes through the 3 image points that I had already found -- and not just some arbitrary curve that passes through the 3 image points. Thanks, $\endgroup$ – User001 Aug 9 '15 at 2:33
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    $\begingroup$ Right, @LebronJames, the fact that Möbius transformations preserve circles (a straight line is a circle passing through $\infty$) settles it. $\endgroup$ – Daniel Fischer Aug 9 '15 at 7:34

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