Asymptotic Nets and Discrete Affine Surfaces with Indefinite Metric

TL;DR

This paper develops a framework for associating affine discrete geometric concepts to non-degenerate asymptotic nets, including area, curvature, and cubic form, with applications to affine minimal surfaces and spheres.

Contribution

It introduces a comprehensive set of affine geometric concepts for asymptotic nets and establishes their structural and compatibility relations, extending discrete differential geometry.

Findings

Defined discrete affine area, mean curvature, and cubic form for asymptotic nets

Established structural and compatibility equations linking these concepts

Analyzed special cases of affine minimal surfaces and affine spheres

Abstract

Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean curvature, normal and co-normal vector fields and cubic form, and they are related by structural and compatibility equations. We consider also the particular cases of affine minimal surfaces and affine spheres.