I am given four points on a straight line, which are mapped to four points on a circle. I confirmed that the latter four points are concyclic by calculating their cross ratio, which is real.
Does a Möbius transformation preserve the order of the points on a straight line when they are mapped to a circle? If so, is it because these transformations preserve orientation?
The question I am looking at is asking me why there is no Möbius transformation with the property that it maps each of the four points on the line to each of the four points on the circle respectively, and I can see that the points on the circle are not in the same order, going around the circle, as the points on the straight line that are mapped to each of them. Or is there more to it?
PS I can also show that the system of equations does not exist, by forming four equations and substituting in the points, but this seems like a lot of work!
I have just realised that the cross ratio of my original four points is different than that of the four points they are mapped to. Since Möbius transformations preserve cross ratios, the suggested mapping is not a Möbius transformation.