Discretization of asymptotic line parametrizations using hyperboloid surface patches
Discretization of asymptotic line parametrizations using hyperboloid surface patches
复制标题
使用双曲面曲面片对渐近线参数化进行离散化
DOI：
10.1007/s10711-013-9830-9
复制
发表时间：
2011
影响因子：
0.5
通讯作者：
Thilo Rörig
中科院分区：
文献类型：
--
作者：
Emanuel Huhnen;Thilo Rörig
文档类[12pt]{minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsFonts}\usepackage{amssymb}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemarin}{-69pt}\Begin{Document}$$3$$\end{Document}-空间中的二维仿射A-网是四边形网格，其离散曲面沿渐近直线参数化。A-网的定义性质是顶点星的平面性，因此一般A-网的初等四边形是倾斜的。本文通过将双曲面曲面片粘合到斜四边形上，将A网推广到可微曲面上。所得到的曲面称为“双曲网”，是一种新的、沿渐近直线参数化的曲面的分段光滑离散化。如果所有的四边形都是等扭的，则单连通仿射A网可以扩展为双曲网。等扭的几何条件隐含着A网的所有内部顶点必须是偶次的组合性质。如果A-网可以扩展到双曲网，则存在这样的扩展的一个参数族。简要说明了如何在计算机上实现双曲网的生成。我们使用Plücker线几何的射影模型来描述A网和双曲面。
Two-dimensional affine A-nets in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3$$\end{document}-space are quadrilateral meshes that discretize surfaces parametrized along asymptotic lines. The defining property of A-nets is planarity of vertex stars, hence elementary quadrilaterals of a generic A-net are skew. The present article deals with the extension of A-nets to differentiable surfaces, by gluing hyperboloid surface patches into the skew quadrilaterals. The obtained surfaces, named “hyperbolic nets”, are a novel, piecewise smooth discretization of surfaces parametrized along asymptotic lines. A simply connected affine A-net can be extended to a hyperbolic net if all quadrilateral strips are “equi-twisted”. The geometric condition of equi-twist implies the combinatorial property, that all inner vertices of the A-net have to be of even degree. If an A-net can be extended to a hyperbolic net, then there exists a 1-parameter family of such extensions. It is briefly explained how the generation of hyperbolic nets can be implemented on a computer. We use a projective model of Plücker line geometry in order to describe A-nets and hyperboloids.