Rigidity of Linearly Constrained Frameworks - Cruickshank, J.; Guler, H.; Jackson, B.; Nixon, A. (Oxford University Press (OUP), 2018)
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 Streinu and Theran [14] in 2010. We will extend their characterisation to the case when $d\geq 3$ and each vertex is constrained to lie in an affine subspace of dimension $t$, when $t=1,2$ and also when $t\geq 3$ and $d\geq t(t-1)$. We then point out that results on body–bar frameworks obtained by Katoh and Tanigawa [8] in 2013 can be used to characterise when a graph has a rigid realisation as a $d$-dimensional body–bar framework with a given set of linear constraints.
Keywords: Mathematics - Metric Geometry; FOS: Mathematics; Mathematics - Combinatorics; Metric Geometry (math.MG); Combinatorics (math.CO); C25, 05C10, 53A05; 16. Peace & justice