曲面三角剖分--度量方法
Surface Triangulation -- The Metric Approach
摘要：
我们开始在一个程序中研究的问题，更好地逼近曲面的三角剖分（三角网格），考虑作为有限度量空间和目标光滑曲面作为其Haussdorff-Gromov极限的近似三角剖分。这使我们能够以更自然的方式定义相关的元素，常量和不变量s.a.主方向和主值、高斯和平均曲率等。通过“自然方式”，我们指的是内在的、离散的、度量的定义，而不是近似或解释可微的概念。通过这种方式，我们希望避免计算错误，事实上，概念的，往往是固有的经典，“数值”的方法。在这第一项研究中，我们考虑的问题，确定一个多面体曲面的高斯曲率，通过使用{ \em嵌入曲率}的意义上的沃尔德（和门格尔）。我们提出了两种方式采用这些定义的高斯曲率的计算。
英文摘要：
We embark in a program of studying the problem of better approximating surfaces by triangulations(triangular meshes) by considering the approximating triangulations as finite metric spaces and the target smooth surface as their Haussdorff-Gromov limit. This allows us to define in a more natural way the relevant elements, constants and invariants s.a. principal directions and principal values, Gaussian and Mean curvature, etc. By a "natural way" we mean an intrinsic, discrete, metric definitions as opposed to approximating or paraphrasing the differentiable notions. In this way we hope to circumvent computational errors and, indeed, conceptual ones, that are often inherent to the classical, "numerical" approach. In this first study we consider the problem of determining the Gaussian curvature of a polyhedral surface, by using the {\em embedding curvature} in the sense of Wald (and Menger). We present two modalities of employing these definitions for the computation of Gaussian curvature.
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