On Weingarten transformations of hyperbolic nets

TL;DR

This paper introduces a canonical discrete analogue of Weingarten transformations for hyperbolic nets, bridging smooth and discrete surface theories, and analyzes their geometric and algebraic properties.

Contribution

It proposes a new discrete Weingarten transformation for hyperbolic nets and proves the existence of such pairs, extending classical concepts to hybrid surface models.

Findings

Existence of Weingarten pairs for hyperbolic nets.

Analysis of geometric properties of the transformations.

Algebraic characterization of the transformations.

Abstract

Weingarten transformations which, by definition, preserve the asymptotic lines on smooth surfaces have been studied extensively in classical differential geometry and also play an important role in connection with the modern geometric theory of integrable systems. Their natural discrete analogues have been investigated in great detail in the area of (integrable) discrete differential geometry and can be traced back at least to the early 1950s. Here, we propose a canonical analogue of (discrete) Weingarten transformations for hyperbolic nets, that is, C^1-surfaces which constitute hybrids of smooth and discrete surfaces "parametrized" in terms of asymptotic coordinates. We prove the existence of Weingarten pairs and analyse their geometric and algebraic properties.