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

An Algebraic Construction of Hyperbolic Planes over a Euclidean Ordered Field