Abstract
Knüppel (J Geom 105:13–20, 2014) shows that in a hyperbolic plane an asymptotic n-gon does not admit a transversal line when n is odd. If n is even then there exist n-gons which admit a transversal line. Recently Struve (J Geom 103:333–346, 2012) showed that in hyperbolic geometry an order structure can be introduced purely in terms of the calculus of reflections. Following this approach we give a short proof of the above mentioned theorem which provides new insights into the underlying foundations of the theorem.
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References
Bachmann, F.: Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd edn. Springer, Heidelberg (1973)
Bachmann, F., Schmidt, E.: n-gons. Math. Expositions, vol. 18. Univ. of Toronto Press, Toronto
Knüppel F.: Transversals in plane hyperbolic geometry. J. Geom. 105, 13–20 (2014)
Struve R.: The calculus of reflections and the order relation in hyperbolic geometry. J. Geom. 103, 333–346 (2012)
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Struve, R. A Note on Transversals in Hyperbolic Planes. Results. Math. 66, 363–366 (2014). https://doi.org/10.1007/s00025-014-0381-7
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DOI: https://doi.org/10.1007/s00025-014-0381-7