Moroccan Journal of Algebra and Geometry with Applications
On a realization theorem in plane hyperbolic geometry and a
related identity in neutral geometry
David E. Dobbs
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320
Pages 361-379 | Received 10 May 2022, Accepted 25 July 2022, Published 06 December 2022
Abstract
Working in the upper half-plane model of plane hyperbolic geometry, we give a new proof that if \alpha \leq \beta \leq \gamma are positive real numbers such that \alpha + \beta + \gamma < \pi, then there exists a hyperbolic triangle whose three (interior) angles haveradian measures \alpha ,\ \beta and \gamma , respectively. Seeking yet another proof of this realization theorem produces a new identity involving the sines and cosines of the angles of any triangle in hyperbolic or Euclidean geometry. The only prerequisites assumed here are some topics in analytic geometry and trigonometry that are typically covered in a precalculus courseand some basic facts from a first course on differential calculus. Thus, much of this paper could be used as enrichment material for a precalculus course or a calculus course, while all of this paper could to used to enrich a course on the classical geometries that features the upper half-plane model.
Keywords: Hyperbolic plane geometry, upper half-plane model, directed angle, tangent line, neutral geometry, identity
MSC numbers: Primary 51-02; Secondary 33B10, 51N20.
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