Global rigidity of 2-dimensional direction-length frameworks with connected rigidity matroids

TL;DR

This paper characterizes when generic 2D direction-length frameworks with connected rigidity matroids are globally rigid, showing they are so if and only if all 2-separations are direction-balanced, advancing understanding of geometric rigidity.

Contribution

It provides a complete characterization of global rigidity for a class of 2D frameworks based on graph separations and direction-length constraints, which was previously unresolved.

Findings

Frameworks are globally rigid if all 2-separations are direction-balanced.

Connected rigidity matroids are key to the characterization.

The result applies to generic frameworks in the plane.

Abstract

A two-dimensional direction-length framework $(G,p)$ consists of a multigraph $G=(V;D,L)$ whose edge set is formed of "direction" edges $D$ and "length" edges $L$, and a realisation $p$ of this graph in the plane. The edges of the framework represent geometric constraints: length edges fix the distance between their endvertices, whereas direction edges specify the gradient of the line through both endvertices. A direction-length framework $(G,p)$ is globally rigid if every framework $(G,q)$ which satisfies the same direction and length constraints as $(G,p)$ can be obtained by translating $(G,p)$ in the plane, and/or rotating $(G,p)$ by $180^{\circ}$. In this paper, we characterise global rigidity for generic direction-length frameworks whose associated rigidity matroid is connected, by showing that such frameworks are globally rigid if and only if every 2-separation of the underlying graph is direction-balanced.