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：每个内蕴多面体曲面都可以通过portalgon表示：即欧几里得平面中多边形的集合，其中某些等长边在抽象意义上被等同。虽然这种表示方法比网格（在R3中形成曲面的平面多边形）更简单，但它具有无界遍历性：曲面中的最短路径可能任意多次访问同一多边形。这种病态行为是高效算法设计的障碍。另一方面，Löffler、Ophelders、Staals和Silveira（SoCG 2023）最近证明了（内蕴）Delaunay三角剖分具有有界遍历性。本文针对由三角形portalgon T表示的闭多面体曲面S，提出一种计算S的Delaunay三角剖分的算法，其顶点为S的奇点（周围角度不等于2π的点）。我们算法的时间复杂度在三角形数量与T的纵横比r的对数上呈多项式关系。在我们的计算模型中，我们证明了对log(r)的依赖是不可避免的。该算法可用于在计算三角形portalgon表面最短路径前进行预处理，并判断两个三角形portalgon的曲面是否等距。