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The calculus of reflections and the order relation in hyperbolic geometry

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Abstract

It is well known that the calculus of reflections (developed by Hjelmslev, Bachmann et al.) enables the derivation of a large part of Euclidean and non-Euclidean geometry without using assumptions about order and continuity. We show in this article that the calculus of reflections can conversely be used to introduce a relation of order in hyperbolic geometry. Our investigations are based on the famous ‘Endenrechnung’ of Hilbert which was formulated purely in terms of the calculus of reflections by F. Bachmann. We then discuss some implications of these results and show that the calculus of reflections enables (1) the introduction of an order relation in a Pappian projective line and (2) to define an axiom system for hyperbolic planes which seems to be as simple as the famous axiom system of Menger who only used the notion of point-line incidence to axiomatize plane hyperbolic geometry.

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  1. SIGNAL IDUNA Gruppe, Joseph-Scherer-Strasse 3, 44139, Dortmund, Germany

    Rolf Struve

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  1. Rolf Struve
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Struve, R. The calculus of reflections and the order relation in hyperbolic geometry. J. Geom. 103, 333–346 (2012). https://doi.org/10.1007/s00022-012-0123-5

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  • DOI: https://doi.org/10.1007/s00022-012-0123-5

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