Global rigidity of 2-dimensional direction-length frameworks

TL;DR

This paper characterizes the conditions under which a generic 2D direction-length framework is globally rigid, reducing the problem to special 'direction irreducible' graphs and establishing necessary and sufficient conditions.

Contribution

It introduces a reduction to direction irreducible graphs and provides a complete combinatorial characterization of global rigidity for generic frameworks.

Findings

Global rigidity is characterized by 2-connectivity, direction-balance, and redundant rigidity.

Reduction to direction irreducible mixed graphs simplifies the analysis.

Provides necessary and sufficient conditions for generic global rigidity.

Abstract

A 2-dimensional direction-length framework is a collection of points in the plane which are linked by pairwise constraints that fix the direction or length of the line segments joining certain pairs of points. We represent it as a pair $(G,p)$, where $G=(V;D,L)$ is a `mixed' graph and $p:V\to{\mathbb R}^2$ is a point configuration for $V$. It is globally rigid if every direction-length framework $(G,q)$ which satisfies the same constraints can be obtained from $(G,p)$ by a translation or a rotation by $180^\circ$. We show that the problem of characterising when a generic framework $(G,p)$ is globally rigid can be reduced to the case when $G$ belongs to a special family of `direction irreducible' mixed graphs, and prove that {every} generic realisation of a direction irreducible mixed graph $G$ is globally rigid if and only if $G$ is 2-connected, direction-balanced and redundantly rigid.