Rigidity of Linearly Constrained Frameworks

We consider the problem of characterising the generic rigidity of bar-joint frameworks in 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≥q 3 and each vertex is constrained to lie in an affine subspace of dimension t, when t=1,2 and also when t≥q 3 and d≥q 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.