New Foundations of Absolute Geometry

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This chapter discusses the absolute geometry that is treated as a common part of Euclidean, including parabolic and hyperbolic geometries. Absolute geometry can also be treated as a common part of three geometries, such as parabolic, hyperbolic, and elliptic. The problem of a uniform description of all models of these three theories is still open. The character of the whole discussion in the chapter seems to be purely geometrical. Although in reality only the form is geometrical; the content is meta-geometrical. For constructing an algebraic structure an ordered field in geometry is a geometrical problem. In meta-geometrical language the chapter reaches at the representation theorem for absolute geometry—that is, at a uniform description of all models for both Euclidean geometry and hyperbolic geometry.