Discrete Gradient Line Fields on Surfaces

TL;DR

This paper introduces a discretization method for Morse-Smale line fields on surfaces, extending discrete vector field techniques to line fields, with applications in geometry and physics.

Contribution

It proposes a novel discretization of Morse-Smale line fields on surfaces, generalizing Forman's discrete vector field construction and defining critical elements via local matchings.

Findings

Euler theorem and homotopy characterization hold for the discretized line fields.

The method extends discrete vector field techniques to line fields on surfaces.

Applications include modeling curvature directions and stress flux in physical systems.

Abstract

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.