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, Löffler, 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.

翻译：每个内在多面体的表面都可以用一个"门多边形"表示：即欧几里得平面中一组多边形，其中某些等长边对以抽象方式标识。尽管这种表示在直觉上比网格（R³中构成表面的平面多边形）更简单，但它存在"无界快乐性"：表面上的最短路径可能会任意多次经过同一个多边形。这种病态行为成为高效算法的障碍。另一方面，Löffler、Ophelders、Staals和Silveira（SoCG 2023）近期证明（内在）德劳内三角剖分具有有界快乐性。本文针对由三角形门多边形T表示的闭多面体表面S，提出一种算法来计算S的德劳内三角剖分，其顶点为S的奇异点（周围角度不等于2π的点）。我们算法的时间复杂度关于三角形数量以及T的纵横比r的对数呈多项式级。在我们的计算模型内，我们证明对log(r)的依赖是不可避免的。该算法可用于在计算表面最短路径前预处理三角形门多边形，以及判断两个三角形门多边形的表面是否等距。