曲面上的离散梯度线场
Discrete Gradient Line Fields on Surfaces
摘要：
流形上的线场是一个光滑映射，它给流形上除有限个点外的所有点都指定一条切线。因此，它可以被看作是向量场的推广。它们模拟了许多几何和物理特性，例如表面上的主曲率方向动力学或弹性中的应力通量。本文提出了曲面上Morse-Smale线场的离散化方法，推广了Forman关于离散向量场的构造。更一般的临界元及其指标是由局部匹配定义的，欧拉定理和同伦型的临界胞的特征仍然成立。
英文摘要：
A line field on a manifold is a smooth map which assigns a tangent line to all but a finite number of points of the manifold. As such, it can be seen as a generalization of vector fields. They model a number of geometric and physical properties, e.g. the principal curvature directions dynamics on surfaces or the stress flux in elasticity. We propose a discretization of a Morse-Smale line field on surfaces, extending Forman's construction for discrete vector fields. More general critical elements and their indices are defined from local matchings, for which Euler theorem and the characterization of homotopy type in terms of critical cells still hold.
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