Approximating geodesics of hyperbolic type metrics on planar domains

TL;DR

This paper introduces an algorithm based on Dijkstra's method to approximate geodesics and distances in planar domains with hyperbolic type metrics, with applications to quasihyperbolic metrics in polygonal domains.

Contribution

It presents a novel algorithm for approximating shortest geodesics and distances in hyperbolic type metrics on planar domains, including polygonal domains, and analyzes its accuracy.

Findings

The algorithm effectively approximates shortest distances in polygonal domains.

Experimental results confirm theoretical properties of geodesics, such as the hyperbolic-quasihyperbolic distance relationship.

The study explores bifurcation phenomena of geodesics and their connection to the domain's medial axis.

Abstract

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.