Shortest paths in planar domains with hyperbolic type metrics
For researchers in computational geometry and geometric group theory, this provides a practical method for approximating geodesics in hyperbolic-type metrics, though the contribution is incremental.
The paper presents an algorithm based on Dijkstra's algorithm to approximate shortest distances in polygonal domains under the quasihyperbolic metric, and experimentally validates theoretical properties of geodesics such as bifurcation and medial axis connections.
We study planar domains $G$ equipped with a hyperbolic type metric and approximate geodesics that join two points $x,y \in G$ and their lengths. We present an algorithm that enables one to approximate the shortest distance in polygonal domains taken with respect to the quasihyperbolic metric. The method is based on Dijkstra's algorithm, and we give several examples demonstrating how the algorithm works and analyze its accuracy. We experimentally demonstrate several previously theoretically observed features of geodesics, such as the relationship between hyperbolic and quasihyperbolic distance in the unit disk. We also investigate bifurcation of geodesics and the connection of this phenomenon to the medial axis of the domain.