Global rigidity of 2-dimensional linearly constrained frameworks

TL;DR

This paper characterizes when generic 2D linearly constrained frameworks are globally rigid, extending previous work, and provides conditions and solution counts for non-balanced cases.

Contribution

It offers an analogous characterization for 2D global rigidity, including conditions for non-balanced frameworks and solution counts based on the rigidity matroid.

Findings

A framework is globally rigid iff it is redundantly rigid and balanced.

Determined the number of solutions for non-balanced frameworks with connected rigidity matroid.

Provided stress matrix and Hendrickson-type necessary conditions for global rigidity.

Abstract

A linearly constrained framework in $\mathbb{R}^d$ is a point configuration together with a system of constraints which fixes the distances between some pairs of points and additionally restricts some of the points to lie in given affine subspaces. It is globally rigid if the configuration is uniquely defined by the constraint system, and is rigid if it is uniquely defined within some small open neighbourhood. Streinu and Theran characterised generic rigidity of linearly constrained frameworks in $\mathbb{R}^2$ in 2010. We obtain an analagous characterisation for generic global rigidity in $\mathbb{R}^2$. More precisely we show that a generic linearly constrained framework in $\mathbb{R}^2$ is globally rigid if and only if it is redundantly rigid and `balanced'. For generic frameworks which are not balanced, we determine the precise number of solutions to the constraint system whenever the underlying rigidity matroid of the given framework is connected. We also obtain a stress matrix sufficient condition and a Hendrickson type necessary condition for a generic linearly constrained framework to be globally rigid in $\mathbb{R}^d$.