Inversive geometry

Summary

Inversive geometry studies the plane with one point at infinity added. It keeps the properties of figures that are preserved by inversions in circles and reflections in lines. In it, lines count as circles through the point at infinity, and inversions map circles to circles. Its transformations are the Möbius transformations and their composites with complex conjugation. So it is the geometry of the Riemann sphere in the sense of the Erlangen program.