On Generic Linearly Constrained Frameworks

A linearly constrained framework in $\mathbb{R}^d$ is a bar-joint framework where, in addition, vertices with loops are constrained to lie in given affine subspaces. In the generic case, when each vertex is incident to sufficiently many loops, a characterisation of rigidity was obtained by Jackson, Nixon and Tanigawa for all $d\geq 3$. By extending this to characterise the rank function of the linearly constrained rigidity matroid (under the same loop hypothesis), sufficient conditions for a looped simple graph to be (globally) rigid in $\mathbb{R}^d$ are obtained. In the 2-dimensional case generic rigidity was characterised by Streinu and Theran, and we obtain a sharper sufficient condition in this case. A key technique is the application of the discharging method.