We know that Mobius transformations $z\to\frac{az+b}{cz+d}$ permutes circles and lines in the Euclidean plane, $(\mathbb{R}^2, dx^2 + dy^2 )$.  

It may even be possible to write an explicit formula for the general Mobius action on any given circle:

$$ |z-z_0|=r \mapsto \left|\frac{az+b}{cz+d} -z_0 \right|=r$$

Such a space can be generated by translations $z \mapsto z + z_1$, rotations $z \mapsto \omega z$, dilations $z \mapsto rz$ and inversions $\displaystyle z = \frac{1}{\overline{z}}$.  The action on all circles is clear except for the last case:

$$ |z-z_0|=r \mapsto \left| z -  \frac{\tfrac{1}{2}z_0}{|z_0|^2 - r^2} \right|= \frac{\tfrac{1}{2}r}{|z_0|^2 - r^2} $$

Is there more succinct way to write this transformation as a Lie group action? 

Here, the [Mobius group](https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Lorentz_transformations) $G = PSL(2, \mathbb{C}) \simeq SO(1,3)^+$ should act on on the space of circles (and lines) in the plane $\mathbb{R}^2$.

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This question also leads me to wonder what the space of circles in the Euclidean plane should be.  Naively, the circles are a copy of $\mathbb{C} \times \mathbb{R}^+$ which looks like it could possibly be [Hyperbolic space](https://en.wikipedia.org/wiki/Hyperbolic_space) $\mathbb{H}^3$, in which case there would even be a natural metric.

However, the lines all have infinite radius.  In which case, we still have 3 parameters, now a copy of $\mathbb{C} \times S^1$ identifying each line with the closest point to the origin.  The $S^1$ behaving something like the [circle at infinity](https://www.youtube.com/watch?v=k-uWXIB7W7Y) (Youtube).

So now my space of circles is $\big(\mathbb{C}\times \mathbb{R}^+ \big) \cup \big(\mathbb{C}\times S^1 \big)$.  How does that happen?