Global Rigidity of 2D Linearly Constrained Frameworks

Abstract

AbstractA linearly constrained framework in $\mathbb{R}^d$ is a point configuration together with a system of constraints that 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. 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 unbalanced generic frameworks, we determine the precise number of solutions to the constraint system whenever the rigidity matroid of the framework is connected. We obtain a stress matrix sufficient condition and a Hendrickson type necessary condition for a generic linearly constrained framework to be globally rigid in $\mathbb{R}^d$.