Editing Inversive geometry (section)

== Axiomatics and generalization ==
One of the first to consider foundations of inversive geometry was [[Mario Pieri]] in 1911 and 1912.<ref>M. Pieri (1911,12) "Nuovi principia di geometria della inversion", ''Giornal di Matematiche di Battaglini'' 49:49&ndash;96 & 50:106&ndash;140</ref> [[Edward Kasner]] wrote his thesis on "[[Invariant theory]] of the inversion group".<ref>{{cite journal | jstor = 1986367 | pages = 430–498 | last1 = Kasner | first1 = E. | title = The Invariant Theory of the Inversion Group: Geometry Upon a Quadric Surface | volume = 1 | issue = 4 | journal = [[Transactions of the American Mathematical Society]] | year = 1900 | doi = 10.1090/S0002-9947-1900-1500550-1 | hdl = 2027/miun.abv0510.0001.001 | hdl-access = free }}</ref>

More recently the [[mathematical structure]] of inversive geometry has been interpreted as an [[incidence structure]] where the generalized circles are called "blocks": In [[incidence geometry]], any [[affine plane (incidence geometry)|affine plane]] together with a single [[point at infinity]] forms a [[Möbius plane]], also known as an ''inversive plane''. The point at infinity is added to all the lines. These Möbius planes can be described axiomatically and exist in both finite and infinite versions.

A [[Model (model theory)|model]] for the Möbius plane that comes from the Euclidean plane is the [[Riemann sphere]].