On constructing angles with prescribed vertex and measure
in the upper half-plane model of hyperbolic geometry

Abstract

It is proved that if k,m \in \mathbb{R} and P_0 \, (x_0,y_0) is a point in the Euclidean plane \mathbb{R}^2 with y_0 \neq k and with \mathcal{X} denoting the (horizontal) line with Cartesian equation y=k , then there exists a unique circle, say \mathcal{K}, such that the center of \mathcal{K} is on \mathcal{X}, \ P_0 lies on \mathcal{K}, and the tangent to \mathcal{K} at P_0 has slope m. An ensuing multi-step algorithmic result that is proved here for the Euclidean upper half-plane determines the angular measure of any directed angle that is formed by counterclockwise rotation from a designated initial side to a designated terminal side, in case each of those “sides” is a hyperbolic line segment (that is, either a vertical (Euclidean) line segment or an arc of a (Euclidean) circle centered on the x -axis). One consequence (for the Euclidean upper half-plane) is the construction of (the unique hyperbolic line segment playing the role of terminal (resp., initial) side of) a unique directed angle having a prescribed vertex, a prescribed measure between 0 and \pi , and a prescribed hyperbolic line segment as initial (resp., terminal) side. As the only prerequisites assumed here are related topics in analytic geometry and trigonometry that can be covered in a precalculus course, this paper could be used as enrichment material for a precalculus course, a calculus course, or a course on the classical geometries that features the upper half-plane model of hyperbolic plane geometry.

Keywords:  Hyperbolic plane geometry, upper half-plane model, directed angle, vertex, Euclidean geometry, slope, tangent line, inverse tangent function, angle of inclination, inverse cosine function, bowedgeodesic, straight geodesic.

MSC numbers: Primary 51-02; Secondary 33B10, 51N20, 51M04.

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