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!
Best Answer
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,
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).