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.