String-node nets and meshes

TL;DR

This paper introduces string-node nets and meshes, new geometric structures with distance constraints, explores their construction, classification, and rigidity properties, advancing understanding of their geometric and mechanical behaviors.

Contribution

It presents the concept of string-node nets and meshes, provides construction methods, and classifies linear meshes, highlighting their rigidity and flexing properties.

Findings

Defined and classified regular linear meshes in R^2 and R^3

Analyzed rigidity and flexing properties of fundamental meshes

Provided construction schemes including minimal extensions

Abstract

New classes of distance-constrained structures are introduced, namely string-node nets and meshes, a mesh being a string-node net for which the nodes are dense in the strings. Various construction schemes are given including the minimal extension of a (countable) line segment net by an arbitrary countable scaling group. A linear mesh has strings that are straight lines and nodes given by the intersection points of these lines. Classes of linear meshes, such as the regular linear meshes in R^2 and R^3, are defined and classified. Contrasting flexing and rigidity properties are obtained for some fundamental meshes in the plane with respect to noncrossing motions in the space of smooth meshes.