The geometry of discrete asymptotic-geodesic 4-webs in isotropic 3-space

Christian Müller, Helmut Pottmann
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Abstract

Abstract The geometry of webs has been investigated over more than a century driven by still open problems. In our paper we contribute to extending the knowledge on webs from the perspective of the geometry of webs on surfaces in three dimensional space. Our study of AGAG-webs is motivated by architectural applications of gridshell structures where four families of manufactured curves on a curved surface are realizations of asymptotic lines and geodesic lines. We describe all discrete AGAG-webs in isotropic space and propose a method to construct them. Furthermore, we prove that some sub-nets of an AGAG-web are timelike minimal surfaces in Minkowski space and can be embedded into a one-parameter family of discrete isotropic Voss nets.

Abstract Image

三维各向同性空间中离散渐近测地线四网的几何特性
一个多世纪以来,人们一直在研究腹网的几何结构,但仍存在一些悬而未决的问题。在我们的论文中,我们从三维空间表面上腹板的几何角度扩展了关于腹板的知识。我们对agag网的研究是由网格壳结构的建筑应用驱动的,在网格壳结构中,曲面上的四族人造曲线是渐近线和测地线的实现。本文描述了各向同性空间中所有离散agag网络,并提出了一种构造agag网络的方法。此外,我们证明了AGAG-web的一些子网络是Minkowski空间中的类时最小曲面,并且可以嵌入到一个单参数离散各向同性Voss网络族中。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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