How to recognize if a Möbius transformation includes an 'inversion' operation?

Question

For my current project, I am looking into Möbius transformations from the unit disk onto itself. Such Möbius transformations can be found, for example, by specifying three points and their images on the unit circle. Assume that I have the four parameters $a$, $b$, $c$ and $d$ of a given Möbius transformation:

$$M(z) = \frac{az+b}{cz+d}$$

Sometimes, the Möbius transformation causes an inversion, i.e. a mapping from the inside of the unit disk to the outside of the unit disk and vice versa.

Now my question: Is there a way to recognize whether a Möbius transformation induces an inversion directly through the parameters $a$, $b$, $c$, and $d$?

Answer 1

If the Möbius transformation maps $\frac12+\frac12i$ outside the unit disc, then it indicates that the transformation is an inversion. Now $$|M(\tfrac12+\tfrac12i)|^2 = \frac{(a+b)^2 + b^2}{(c+d)^2+d^2},$$ and when this is $>1$, you have that $\frac12+\frac12i$ is mapped outside the disc. Thus you could take your condition to be that $$(a+b)^2+b^2>(c+d)^2+d^2.$$

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Edit: As suggested in runaway44's comment, we can obtain an easier condition by seeing where $0$ is mapped. Indeed, $$|M(0)| = b/d,$$ so we just need $|b| > |d|$.