Leírás
A bar-joint framework $(G,p)$ is the combination of a finite simple graph
$G=(V,E)$ and a map $p:V\rightarrow \mathbb{R}^d$. A bar-joint framework is
rigid in $\mathbb{R}^d$ if the only edge-length preserving continuous
motions of the vertices arise from isometries of $\mathbb{R}^d$. It is well
known that, generically, rigidity of such frameworks can be understood in
purely combinatorial terms when $d=1,2$ and finding such a
characterisation is an open problem when $d\geq 3$. 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 in 2010. We will extend their characterisation to the
case when $d\geq 3$ and each vertex is constrained to lie on an affine
subspace of dimension at most two. By exploiting a natural correspondence
with frameworks whose vertices are constrained to lie on a surface, we also
characterise generic rigidity for frameworks on a generic surface in
$\mathbb{R}^d$.
Joint work with Hakan Guler and Bill Jackson