Infinitesimal rigidity for non-Euclidean bar-joint frameworks

TL;DR

This paper characterizes the minimal infinitesimal rigidity of bar-joint frameworks in non-Euclidean spaces using (2,2)-tight graphs, extending rigidity theory beyond Euclidean norms.

Contribution

It provides a combinatorial characterization of infinitesimal rigidity for frameworks in non-Euclidean plane spaces based on graph properties.

Findings

Rigidity characterized by (2,2)-tight graphs

Frameworks are minimally infinitesimally rigid if and only if graph conditions are met

Extends rigidity theory to non-Euclidean norms

Abstract

The minimal infinitesimal rigidity of bar-joint frameworks in the non-Euclidean spaces (R^2, ||.||_q) are characterised in terms of (2,2)-tight graphs. Specifically, a generically placed bar-joint framework (G,p) in the plane is minimally infinitesimally rigid with respect to a non-Euclidean l^q norm if and only if the underlying graph G = (V,E) contains 2|V|-2 edges and every subgraph H=(V(H),E(H)) contains at most 2|V(H)|-2 edges.