Computing the Intrinsic Delaunay Triangulation of a Closed Polyhedral Surface

TL;DR

This paper presents a polynomial-time algorithm for computing the intrinsic Delaunay triangulation of a closed polyhedral surface represented by a triangular portalgon, facilitating efficient shortest path computations and surface isometry checks.

Contribution

It introduces a novel algorithm for Delaunay triangulation of polyhedral surfaces in portalgon form, with complexity depending on the aspect ratio, and proves the logarithmic dependency is necessary.

Findings

Algorithm runs in polynomial time in number of triangles and log of aspect ratio

Dependency on log(r) in complexity is proven to be unavoidable

Enables pre-processing for shortest path computation and isometry testing

Abstract

Every surface that is intrinsically polyhedral can be represented by a portalgon: a collection of polygons in the Euclidean plane with some pairs of equally long edges abstractly identified. While this representation is arguably simpler than meshes (flat polygons in R3 forming a surface), it has unbounded happiness: a shortest path in the surface may visit the same polygon arbitrarily many times. This pathological behavior is an obstacle towards efficient algorithms. On the other hand, L\"offler, Ophelders, Staals, and Silveira (SoCG 2023) recently proved that the (intrinsic) Delaunay triangulations have bounded happiness. In this paper, given a closed polyhedral surface S, represented by a triangular portalgon T, we provide an algorithm to compute the Delaunay triangulation of S whose vertices are the singularities of S (the points whose surrounding angle is distinct from 2pi). The time complexity of our algorithm is polynomial in the number of triangles and in the logarithm of the aspect ratio r of T. Within our model of computation, we show that the dependency in log(r) is unavoidable. Our algorithm can be used to pre-process a triangular portalgon before computing shortest paths on its surface, and to determine whether the surfaces of two triangular portalgons are isometric.