Discrete Affine Surfaces based on Quadrangular Meshes

TL;DR

This paper develops a framework for defining affine structures on discrete quadrangular surfaces, introducing two classes with associated invariants and conditions for their existence, advancing discrete differential geometry.

Contribution

It introduces two new classes of discrete affine surfaces based on quadrangular meshes, with associated invariants and existence conditions, expanding the theory of discrete affine differential geometry.

Findings

Defined indefinite and definite discrete affine surfaces

Established invariants such as metrics and curvature for these surfaces

Derived necessary and sufficient conditions for their existence

Abstract

In this paper we are interested in defining affine structures on discrete quadrangular surfaces of the affine three-space. We introduce, in a constructive way, two classes of such surfaces, called respectively indefinite and definite surfaces. The underlying meshes for indefinite surfaces are asymptotic nets satisfying a non-degeneracy condition, while the underlying meshes for definite surfaces are non-degenerate conjugate nets satisfying a certain natural condition. In both cases we associate to any of these nets several discrete affine invariant quantities: a metric, a normal and a co-normal vector fields, and a mean curvature. Moreover, we derive structural and compatibility equations which are shown to be necessary and sufficient conditions for the existence of a discrete quadrangular surface with a given affine structure.