Bachmann's reflection geometry

Summary

Bachmann's reflection geometry builds plane absolute geometry from a group generated by reflections. Lines are replaced by the reflections in them, and points by the half-turns about them. Incidence and perpendicularity become algebraic relations between involutions, the elements of order $2$ of the group. The resulting structures are called metric planes, and they include Euclidean, hyperbolic and elliptic planes over many fields. Friedrich Bachmann developed this approach in his book "Aufbau der Geometrie aus dem Spiegelungsbegriff".