AI 中文总结
本文提出首个从基础多边形表示构造双曲曲面ε-网的算法，引入伪ε-网概念克服短测地线领圈带来的大小界定难题，相关算法可用于计算双曲曲面长度谱与最短非平凡闭测地线长度。
AI 中文摘要
双曲曲面是数学中的基本对象，在计算几何与拓扑学中愈发重要。这类曲面上高效算法设计的关键要素之一，是存在复杂度可控的几何离散化。本文提出了首个从基础多边形表示出发构造双曲曲面上ε-网的算法，该方法基于Delaunay细化，通过边翻转维护Delaunay三角剖分。由于短测地线周围存在任意长的领圈，ε-网的大小无法仅亏格作为函数界定；为克服此难点，引入伪ε-网的概念，将曲面分解为ε-细圆柱，以及剩余厚部分上ε-网对应的Delaunay三角剖分。作为应用，得到计算ε-厚双曲曲面长度谱，以及从伪log(√2)-网计算 systole（最短非平凡闭测地线长度）的算法。这些结果表明，基于Delaunay的离散化为双曲曲面上的算法计算提供了实用且通用的框架。
英文摘要
Hyperbolic surfaces are a fundamental object in mathematics and play an increasingly important role in computational geometry and topology. A key ingredient in the design of efficient algorithms on such surfaces is the availability of a geometric discretization of controlled complexity. In this paper, we present the first algorithm for constructing e-nets on hyperbolic surfaces starting from a fundamental polygon representation. Our approach is based on Delaunay refinement and relies on maintaining Delaunay triangulations through edge flips. The size of an e-net cannot be bounded solely as a function of the genus because of the presence of arbitrarily long collars around short geodesics. To overcome this difficulty, we introduce the notion of a pseudo e-net, which decomposes the surface into e-thin cylinders together with a Delaunay triangulation over an e-net of the remaining thick part. As applications, we obtain algorithms for computing the length spectrum of an e-thick hyperbolic surface and for computing the systole from a pseudo log(sqrt(2))-net. These results demonstrate that Delaunay-based discretizations provide a practical and versatile framework for algorithmic computations on hyperbolic surfaces.
