Local and Infinitesimal Rigidities

Abstract

This chapter focuses on the problems of local rigidity and infinitesimal rigidity of bar frameworks. These problems have a long and rich history going back at least as far as Cauchy [51]. The main tools in tackling these problems are the rigidity matrix R and the dual rigidity matrix \(\bar{R}\). While R is defined in terms of the underlying graph G and configuration p, \(\bar{R}\) is defined in terms of the complement graph \(\bar{G}\) and Gale matrix Z. Nonetheless, both matrices R and \(\bar{R}\) carry the same information. The chapter concludes with a discussion of generic local rigidity in dimension 2, where the local rigidity problem reduces to a purely combinatorial one depending only on graph G. The literature on the theory of local and infinitesimal rigidities is vast [59, 57, 66, 97, 194]. However, in this chapter, we confine ourselves to discussing only the basic results and the results pertaining to EDMs.