论文标题
扁平托里的通用三角剖分
A Universal Triangulation for Flat Tori
论文作者
论文摘要
由于Burago和Zalgaller（1960，1995）引起的结果指出，每个可定向的多面体表面（通过粘合欧几里得多边形获得）都具有等距的分段线性（PL）嵌入到欧几里得空间$ \ Mathbb {e}^3 $中的euclidean空间。由欧几里得平行四边形的相对侧鉴定而产生的平坦圆环，是多面体表面的一个简单例子。在第一部分中，我们适应了Burago和Zalgaller的证明，该证明是部分非构造性的，以产生Flat Tori的PL等距嵌入。我们的实现会产生带有大量顶点的嵌入，而且每个平坦的圆环都不同。在第二部分中，基于Zalgaller（2000）的另一种结构以及Arnoux等人的最新作品。 （2021），我们表现出具有2434个三角形的通用三角剖分，可以在每个三角形上线性嵌入，以实现任何平坦的圆环的度量。
A result due to Burago and Zalgaller (1960, 1995) states that every orientable polyhedral surface, one that is obtained by gluing Euclidean polygons, has an isometric piecewise linear (PL) embedding into Euclidean space $\mathbb{E}^3$. A flat torus, resulting from the identification of the opposite sides of a Euclidean parallelogram, is a simple example of polyhedral surface. In a first part, we adapt the proof of Burago and Zalgaller, which is partially non-constructive, to produce PL isometric embeddings of flat tori. Our implementation produces embeddings with a huge number of vertices, moreover distinct for every flat torus. In a second part, based on another construction of Zalgaller (2000) and on recent works by Arnoux et al. (2021), we exhibit a universal triangulation with 2434 triangles which can be embedded linearly on each triangle in order to realize the metric of any flat torus.
