OPERATIONS PRESERVING GLOBAL RIGIDITY OF GENERIC DIRECTION-LENGTH FRAMEWORKS

Abstract

A two-dimensional direction-length framework is a pair (G, p), where G = (V; D, L) is a graph whose edges are labeled as 'direction' or 'length' edges, and a map p from V to ℝ2. The label of an edge uv represents a direction or length constraint between p(u) and p(v). The framework (G, p) is called globally rigid if every other framework (G, q) in which the direction or length between the endvertices of corresponding edges is the same, is 'congruent' to (G, p), i.e. it can be obtained from (G, p) by a translation and, possibly, a dilation by -1. We show that labeled versions of the two Henneberg operations (0-extension and 1-extension) preserve global rigidity of generic direction-length frameworks. These results, together with appropriate inductive constructions, can be used to verify global rigidity of special families of generic direction-length frameworks.