How do I know if a fractional linear transformation exists?

Question

I have a feeling I'm missing another obvious point about FLTs. How do I know if a specific fractional linear transformation exists? I think I can find specific transformations by using the cross-ratios, but how do I know if such a transformation is even possible? For example how would I know if it's possible to find a transformation that sends circles {z:|z|=1} and {z:|z|=2} to parallel lines or other things of the sort?

Answer 1

Fractional linear transforms have a very useful property called 3-transitivity. If $(z_1, z_2, z_3)$ and $(w_1, w_2, w_3)$ are two sets of distinct points in $\mathbb{C} \cup \infty$, then there is a fractional linear transform between them (i.e. a transform such that $z_i \mapsto w_i$). Furthermore, the transform is unique ; though this means that in general, there is no transform mapping a specific 4th point to another. Somewhat relatedly, if the transform fixes $(z_1, z_2, z_3)$, then it is the identity map.

FLTs also have the property that they map circles to circles (note that in the complex plane, we may think of straight lines as circles through the origin). Since any three distinct points in $\mathbb{C} \cup \infty$ specify a unique circle, we may use 3-transitivity to map any circle to any other circle. Note this mapping is not unique however, since many sets of three points define a particular circle.

In answer to your more specific question, there is no transformation between two parallel lines and two concentric circles. The two circles don't touch anywhere, but the two parallel lines do... at $\infty$! So under any transformation $T$ on the two lines, $T(\infty)$ must be a member of the images of both lines. You can map two parallel lines to a pair of circles like for example $\{z : |z| = 2\}$ and $\{z : |z-1| = 1\}$ though, since in this case they touch only once at $z = 2$.

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(Further information if you find this subject interesting):

FLTs are part of a larger group of transforms, those which are bijective and holomorphic (together, biholomorphic). Further, one may ask whether it is possible to map any subset of the complex plane to any other subset biholomorphically. The answer is yes; this is guarenteed by the Riemann Mapping Theorem: There exists a biholomorphic map between any simply connected strict subset of the complex plane and the unit disc.