A discrete parametrized surface theory in R^3

TL;DR

This paper introduces a comprehensive discrete surface theory in three-dimensional space that unifies various parametrizations, encompassing integrable and non-integrable geometries, and includes constant curvature surface families.

Contribution

It develops a unified discrete surface framework in R^3 that generalizes existing models and incorporates Lax representations and associated constant curvature surface families.

Findings

Unifies multiple discrete surface parametrizations.

Extends to non-integrable geometries.

Includes one-parameter families of constant curvature surfaces.

Abstract

We propose a discrete surface theory in $\mathbb R^3$ that unites the most prevalent versions of discrete special parametrizations. This theory encapsulates a large class of discrete surfaces given by a Lax representation and, in particular, the one-parameter associated families of constant curvature surfaces. The theory is not restricted to integrable geometries, but extends to a general surface theory.