Global Rigidity of 2D Linearly Constrained Frameworks

Abstract

A 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 $\mathbb{R}^d$.

An Improved Bound for the Rigidity of Linearly Constrained Frameworks

“…A linearly constrained frameworks (G, p, q) in R d is globally rigid if it is the only realisation of G in R d which satisfies the same distance and linear constraints as (G, p, q). It was proved in [5] that a linearly constrained framework (G, p, q) in R 2 is globally rigid if and only if every connected component H of G is either a single vertex with at least 2 loops, or is redundantly rigid (i.e. H − f is rigid in R 2 for all edges and loops f of H) and '2-balanced'.…”

“…More precisely we conjecture that, if (G, p, q) is a generic linearly constrained framework with at least two vertices in R d and every vertex of G is incident with at least d 2 loops, then (G, p, q) is globally rigid if and only if every connected component of G is either a single vertex with at least d loops, or is redundantly rigid in R d . (The necessary condition from [5] that G should be 'd-balanced' follows from the assumption that every vertex of G is incident with at least d 2 loops. )…”

Rigidity through a Projective Lens

“…Global rigidity has also been considered for linearly constrained frameworks [170]. In this context natural analogues of Hendrickson's conditions hold and a natural stress matrix condition is sufficient for generic global rigidity.…”

“…It would be interesting to extend these global rigidity results to different types of sliders, as was done for infinitesimal rigidity in [78]. It would also be valuable to generalise the results of [170] to higher dimensions and to allow non-generic linear constraints.…”

Rigidity of symmetric frameworks in normed spaces

“…Streinu and Theran [24] proved a characterisation of generic rigidity for linearly constrained frameworks in R 2 . The articles [5,10] provide an analogous characterisation for generic rigidity of linearly constrained frameworks in R d as long as the dimensions of the affine subspaces at each vertex are sufficiently small (compared to d), and [8] characterises the stronger notion of global rigidity in this context.…”