Global Rigidity of Line Constrained Frameworks

TL;DR

This paper characterizes when bar-joint frameworks with vertices constrained to lines in ^d are globally rigid, extending known 1D results to higher dimensions under mild assumptions.

Contribution

It provides a complete combinatorial characterization of generic global rigidity for line-constrained frameworks in ^d, generalizing 1D results.

Findings

Complete combinatorial characterization of global rigidity in ^d

Extension of 1D global rigidity characterization to higher dimensions

Applicable under mild assumptions on the set of lines

Abstract

We consider the global rigidity problem for bar-joint frameworks where each vertex is constrained to lie on a particular line in $\mathbb R^d$. In our setting we allow multiple vertices to be constrained to the same line. Under a mild assumption on the given set of lines we give a complete combinatorial characterisation of graphs that are generically globally rigid in this setting. This gives a $d$-dimensional extension of the well-known combinatorial characterisation of 1-dimensional global rigidity.