ABSTRACT
Chapter 61 described the basic theory of infinitesimal rigidity of bar and joint structures and a number of related structures. In this chapter, we consider the stronger properties of:
Global Rigidity: given a discrete configuration of points in Euclidean d-space, and a set of fixed pairwise distances, is the set of solutions unique, up to congruence in d-space?
Universal Rigidity: given a discrete configuration of points in Euclidean d-space, and a set of fixed pairwise distances, is the set of solutions unique, up to congruence in all dimensions d ′ ≥ d $ d^{\prime } \ge d $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math63_1.tif"/> ?
How do global rigidity and universal rigidity depend on the combinatorial properties of the associated graph, in which the vertices and edges correspond to points and fixed pairwise distances, respectively, and how do they depend on the specific geometry of the initial configuration?
To study global rigidity, we use vocabulary and techniques drawn from (i) structural engineering: bars and joints, redundant first-order rigidity, static self-stresses (linear techniques); as well as from (ii) minima for energy functions with their companion stress matrices. There are both global rigidity theorems which hold for almost all realizations of a graph G based on combinatorial properties of the graph; and global rigidity theorems that hold for some specific realizations, and depend on the particular details of the geometry of (G, p).