Are the any geometric invariants of special conformal transformations?

Question

So for example, vectors $(x-y)$ are invariant under translations $x\rightarrow x+p$. And distances $|x-y|$ are invariant under translations and rotations $x\rightarrow Mx$.

With three points, angles are invariant under translations, rotations and scales:

$\cos(\theta) = \frac{(x-y).(x-z)}{|x-y||x-z|}$

But are there any geometric quantities involving 2, 3 or 4 points that are invariant also under special conformal transformations? i.e. translations, rotations, scales and inversions $x^\mu\rightarrow x^\mu/|x|^2$ ?

All I know is that circles stay as circles under this transformation so maybe some kind of 'circleness' invariant?

My guess is that it would involve four points and could involve ratios of areas, distances or angles of triangles.

Perhaps there is a non rational invariant? Or one that uses an infinite number of points? (Perhaps there are only invariants based on curves and surfaces and not points?)

Answer 1

I'm not sure I completely understand the question, but perhaps the cross-ratio is what you're looking for?

https://en.wikipedia.org/wiki/Cross-ratio

This thing is an invariant under Mobius transformations, which consist of compositions of translations, dilations (of which rotations are a special case in complex analysis) and inversions, so perhaps it fits. Mobius transformations are conformal on the Riemann sphere - the compactified complex plane.

Answer 2

Under an inversion $x^\mu \rightarrow x^\mu / |x|^2$ then length $|x-y|^2$ becomes $|x/|x|^2-y/|y|^2|^2$ which after rearranging becomes $\frac{|x-y|^2}{|x|^2|y|^2}$. So an invariant on four points is:

$$\frac{ |x-y| |w-z| }{ |x-w| |y-z| }$$