Inversive geometry

Remarks

In the language of the Erlangen program, inversive geometry is the geometry of $\hat{\C}$ with the inversive group as its group of transformations. The elements of the inversive group that preserve orientation are the Möbius transformations $z \mapsto (az + b)/(cz + d)$ with $ad - bc \ne 0$. The others are the maps $z \mapsto (a\bar{z} + b)/(c\bar{z} + d)$ with $ad - bc \ne 0$. Every element maps generalized circles to generalized circles and preserves the size of angles between curves.

Example: the property "these four points lie on a common generalized circle" is inversive, since the inversive group maps generalized circles to generalized circles.

Non-example: the property "this circle has radius $1$" is not inversive; the inversion $z \mapsto 4 / \bar{z}$ maps the circle $\abs{z} = 1$ to the circle $\abs{z} = 4$.

The same construction in $\R^n \cup \set{\infty}$, with inversions in spheres and reflections in hyperplanes, gives inversive geometry in dimension $n$.