Is there a procedural way of finding a Möbius transformation given prescribed conditions?

Question

Is there a procedural way of finding a Möbius transformation given prescribed conditions?

For example, I've been asked to find a Möbius tranformation which fixes $\mathcal{C}_2$, maps $\mathcal{C}_1$ to a line parallel to the imaginary axis, and sends $4$ to $0$. (Here $\mathcal{C}_r$ denotes the complex circle with radius $r$, e.g. $\mathcal{C}_1$ is the unit circle.)

I'm less concerned about solving this particular problem as I am about figuring out if there is a more or less algorithmic way to go about problems of this type. I am aware that a Möbius transformation is completely defined by three points, but I'm not sure how to get this from the set images.

Answer 1

Offhand, this seems like an over-determined problem. But there's some symmetry. Do you know what Möbius transformations map the unit disk to itself? Then can you generalize to the disk of radius $2$? Then think about how to get $4$ to map to $0$.

Answer 2

Usually, the best way to solve these types of problems is by using the cross ratio