Finite and infinitesimal flexibility of semidiscrete surfaces

TL;DR

This paper investigates the flexibility properties of semidiscrete surfaces, establishing conditions for infinitesimal and finite flexibility, and providing differential equations and necessary conditions for various ribbon configurations.

Contribution

It introduces a systematic analysis of infinitesimal and finite flexibility in semidiscrete surfaces, including differential equations and necessary conditions for n-ribbon surfaces.

Findings

2-ribbon surfaces have one degree of flexibility

Derived differential equations for isometric deformations

Every n-ribbon surface has at most one degree of flexibility

Abstract

In this paper we study infinitesimal and finite flexibility for generic semidiscrete surfaces. We prove that generic 2-ribbon semidiscrete surfaces have one degree of infinitesimal and finite flexibility. In particular we write down a system of differential equations describing isometric deformations in the case of existence. Further we find a necessary condition of 3-ribbon infinitesimal flexibility. For an arbitrary $n\ge 3$ we prove that every generic $n$-ribbon surface has at most one degree of finite/infinitesimal flexibility. Finally, we discuss the relation between general semidiscrete surface flexibility and 3-ribbon subsurface flexibility. We conclude this paper with one surprising property of isometric deformations of developable semidiscrete surfaces.