Shortest paths in planar domains with hyperbolic type metrics

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.