Chapter V - Axiomatic Geometry

Abstract

This chapter describes the different axioms of axiomatic geometry. An inversion in a circle is a map of the plane that interchanges points inside and outside of the circle excluding the center. It is the circular analogue of a reflection in a line. Absolute geometry is the study of the properties common to both Euclidean and hyperbolic geometry. The axioms of absolute geometry do not include the Fifth Postulate or its equivalents, and so theorems of absolute geometry apply to both the Euclidean and the hyperbolic planes. The axioms one present are essentially because of George David Birkhoff. They make it possible to derive basic properties quickly and easily by using coordinate systems on lines, and degree measures of angles. The axioms of absolute geometry are used to prove the equivalence of a number of alternative forms of the Fifth Postulate. Each of these equivalent properties characterizes the difference between Euclidean and hyperbolic geometry.