Generalised circles in the context of Moebius transformations

Question

I‘m doing complex analysis at the moment and while speaking about Möbius transformations we introduced the notion of a generalized circle. It seems to be a really important construct while discussing Möbius transformations since this transformations have good properties w.r.t. generalized circles. But somehow I don‘t know how to think about this generalized circles since our Prof. told us they are not circles as in the usual way.

Could maybe someone take some time and explain me how to think about this and also maybe how I get an intuition?

Thanks for your help.

Answer 1

A mobius transformation can be dissected into the following four transformations done successively:

1. Translation
2. Inversion $ \frac{1}{z}$ map
3. Expansion and rotation
4. Another translation

Details in page-124 of Tristan Needham's VCA.

Now, it is pretty clear that step 1.,3.,4. would preserve circle. The only thing in doubt is step 2. . Let us study it in detail.

We can think step-2 as a two-step process:

$$ z \to \frac{1}{\overline{z}} \to \frac{1}{z}$$

Now clearly reflection preserves circles (second step here), so the study boils down first transformation. The first step turns out to be a geometrically rich transformation known as geometric inversion, in particular, we have a geometric inversion over the unit circle. Note that whenever we talk about geometric inversion, we always do it across a circle.

Geometrically speaking, if we were to map the whole plane to a Riemann sphere, the geometric inversion step would just be reflecting the Riemann sphere across the complex plane. Note that the Riemann sphere I am talking about sits like this:

Page-141 , VCA

We can find show that Geometric inversion has the follow properties:

1. If a line $L$ does not pass through the center of the circle of inversion, then inversion maps it to a circle that passes through center. Note that inversion is a self inverse transformation, so it occurs the opposite way too (circle through center-> line not through center). (pg-127)

2. If a circle $C$ does not pass through center of circle of inversion, then inversion maps it another circle not passing through center of circle of inversion.

Furthermore we can see that inversion essentially behaves like reflection along lines as we increase the radius of the circle we are inverting over( Page-130). Here is a gif of how it happens (graph link):

Due to all of this , it would be reasonable here to call a 'line' as a 'circle' as well in the context of mobius transformation.