An improved bound for the rigidity of linearly constrained frameworks

TL;DR

This paper extends the understanding of the rigidity of linearly constrained frameworks in Euclidean space, providing a new bound that broadens the applicability of previous characterisations.

Contribution

It generalizes existing rigidity characterizations to cases where the ambient dimension is at least twice the affine subspace dimension.

Findings

Established a new bound for rigidity in higher dimensions.

Extended previous results to a broader class of frameworks.

Provided a unified framework for understanding rigidity constraints.

Abstract

We consider the problem of characterising the generic rigidity of bar-joint frameworks in $\mathbb{R}^d$ in which each vertex is constrained to lie in a given affine subspace. The special case when $d=2$ was previously solved by I. Streinu and L. Theran in 2010 and the case when each vertex is constrained to lie in an affine subspace of dimension $t$, and $d\geq t(t-1)$ was solved by Cruickshank, Guler and the first two authors in 2019. We extend the latter result by showing that the given characterisation holds whenever $d\geq 2t$.