Double-distance frameworks and mixed sparsity graphs

TL;DR

This paper develops a rigidity theory for frameworks with two types of distance constraints, providing combinatorial characterisations for their rigidity in various metric space contexts.

Contribution

It introduces mixed sparsity graph characterisations for the rigidity of frameworks with dual distance constraints in different geometric settings.

Findings

Characterisations for frameworks on surfaces with Euclidean and geodesic constraints

Rigidity criteria for frameworks in the plane with Euclidean and non-Euclidean distances

Results on direction-length frameworks in non-Euclidean planes

Abstract

A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint frameworks in a variety of such contexts. The main results are combinatorial characterisations for (i) frameworks restricted to surfaces with both Euclidean and geodesic distance constraints, (ii) frameworks in the plane with Euclidean and non-Euclidean distance constraints, and (iii) direction-length frameworks in the non-Euclidean plane.