Axiomatics and Generalization
The Euclidean plane or, more generally, any affine plane together with a single point at infinity added to every line forms a Möbius plane, also known as an inversive plane. These Möbius planes can be described axiomatically and exist in both finite and infinite versions.
A model for the Möbius plane that comes from the Euclidean plane is the Riemann sphere.
Read more about this topic: Inversive Geometry