Geometric properties of hyperbolic geodesics

In the unit disk D hyperbolic geodesic rays emanating from the origin and hyperbolic disks centered at the origin exhibit simple geometric properties. The goal is to determine whether analogs of these geometric properties remain valid for hyperbolic geodesic rays and hyperbolic disks in a simply connected region Ω. According to whether the simply connected region Ω is a subset of the unit disk D, the complex plane C or the extended complex plane (Riemann sphere) C∞ = C∪ {∞}, the geometric properties are measured relative to the background geometry on Ω inherited as a subset of one of these classical geometries, hyperbolic, Euclidean and spherical. In a simply connected hyperbolic region Ω ⊂ C hyperbolic polar coordinates possess global Euclidean properties similar to those of hyperbolic polar coordinates about the origin in the unit disk if and only if the region is Euclidean convex. For example, the Euclidean distance between travelers moving at unit hyperbolic speed along distinct hyperbolic geodesic rays emanating from an arbitrary common initial point is increasing if and only if the region is convex. A simple consequence of this is the fact that the two ends of a hyperbolic geodesic in a convex region cannot be too close. Exact analogs of this Euclidean separating property of hyperbolic geodesic rays hold when Ω lies in either the hyperbolic plane D or the spherical plane C∞.