Frameworks with coordinated edge motions

TL;DR

This paper introduces a rigidity theory for bar-joint frameworks allowing coordinated edge length changes within classes, characterizing rigidity via matroid theory and redundancy in Euclidean spaces.

Contribution

It develops a new rigidity framework for coordinated edge motions and characterizes rigid graphs using matroid union and redundancy concepts.

Findings

Rigidity is a generic property for coordinated frameworks.

Characterization of rigid graphs via redundancy in the rigidity matroid.

Interpretation of results through matroid unions.

Abstract

We develop a rigidity theory for bar-joint frameworks in Euclidean $d$-space in which specified classes of edges are allowed to change length in a coordinated fashion that requires differences of lengths to be preserved within each class. Rigidity for these coordinated frameworks is a generic property, and we characterize the rigid graphs in terms of redundant rigidity in the standard $d$-dimensional rigidity matroid. We also interpret our main results in terms of matroid unions.