Absolute geometry

Remarks

Example: the Euclidean plane $\R^2$ is an absolute plane, since it satisfies all of Hilbert's axioms. The hyperbolic plane is also an absolute plane: it satisfies all these axioms, but through a point off a line there pass several lines that miss it. Non-example: the sphere with great circles as lines is not an absolute plane. Two antipodal points lie on infinitely many great circles, so the incidence axiom "two points lie on exactly one line" fails. With the continuity axioms, every absolute plane is isomorphic to the Euclidean plane or to the hyperbolic plane. The name "neutral geometry" is common in textbooks.