Inversive geometry

Inversive geometry is the geometry of circles and the set of transformations that map all circles into circles. This is an angle-preserving geometry, which is why it is called conformal. For greater than two dimensions, this is also the same as conformal geometry. For two dimensions, however, conformal geometry is simply the Riemann sphere.

Basically, in the spirit of the Erlanger program, inversive geometry is the study of transformations generated by the Euclidean transformations together with inversions, which in coordinate form, basically are conjugate to where r is the radius of the inversion. Note that in inversive geometry, there is no distinction made between a straight line and a circle (or hyperplane and hypersphere): a line is just nothing more and nothing less than a circle in its particular embedding in a Euclidean geometry (with a point added at infinity) and one can always be transformed into another.