An Algebraic Construction of Hyperbolic Planes over a Euclidean Ordered Field

TL;DR

This paper presents an algebraic method to construct hyperbolic planes over Euclidean ordered fields, satisfying key geometric axioms but with hyperbolic parallelism instead of Euclidean.

Contribution

It introduces a novel algebraic construction of hyperbolic planes that adhere to Hilbert's axioms within an algebraic framework over Euclidean ordered fields.

Findings

Constructs hyperbolic planes satisfying Hilbert axioms

Demonstrates hyperbolic parallel axiom in the algebraic setting

Bridges algebraic methods with hyperbolic geometry

Abstract

Using concepts and techniques of bilinear algebra, we construct hyperbolic planes over a euclidean ordered field that satisfy all the Hilbert axioms of incidence, order and congruence for a basic plane geometry, but for which the hyperbolic version of the parallel axiom holds rather than the classical Euclidean parallel postulate.