On the Generating Motions and the Convexity of a Well-Known Curve in Hyperbolic Geometry

In Euclidean geometry the vertices P of those angles ∠APB of size α that pass through the endpoints A, B of a given segment trace the arc of a circle. In hyperbolic geometry on the other hand a set of equivalently defined points P determines a different kind of curve. In this paper the most basic property of the curve, its convexity, is established. No straight-forward proof could be found. The argument rests on a comparison of the rigid motions that map one of the angles ∠APB into other ones.