Shortest Paths in Portalgons
Abstract
Any surface that is intrinsically polyhedral can be represented by a collection of simple polygons (fragments), glued along pairs of equally long oriented edges, where each fragment is endowed with the geodesic metric arising from its Euclidean metric. We refer to such a representation as a portalgon, and we call two portalgons equivalent if the surfaces they represent are isometric. We analyze the complexity of shortest paths in portalgons. We call a fragment happy if any shortest path on the portalgon visits it at most a constant number of times. A portalgon is happy if all of its fragments are happy. We present an efficient algorithm to compute shortest paths on happy portalgons. The number of times that a shortest path visits a fragment is unbounded in general. We contrast this by showing that the intrinsic Delaunay triangulation of any polyhedral surface corresponds to a happy portalgon. Since computing the intrinsic Delaunay triangulation may be inefficient, we provide an efficient algorithm to compute happy portalgons for a restricted class of portalgons.
Keywords
Cite
@article{arxiv.2303.08937,
title = {Shortest Paths in Portalgons},
author = {Maarten Löffler and Tim Ophelders and Frank Staals and Rodrigo I. Silveira},
journal= {arXiv preprint arXiv:2303.08937},
year = {2023}
}
Comments
34 pages. Full version of a paper to appear in a shorter form in the 39th International Symposium on Computational Geometry (SoCG 2023)