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!