On affine motions and bar frameworks in general position

Abstract

A configuration p in r-dimensional Euclidean space is a finite collection of points (p1,…,pn) that affinely span Rr. A bar framework, denoted by G(p), in Rr is a simple graph G on n vertices together with a configuration p in Rr. A given bar framework G(p) is said to be universally rigid if there does not exist another configuration q in any Euclidean space, not obtained from p by a rigid motion, such that ||qi-qj|| = ||pi-pj|| for each edge (i,j) of G.It is known [2,7] that if configuration p is generic and bar framework G(p) in Rr admits a positive semidefinite stress matrix S of rank ( n-r-1), then G(p) is universally rigid. Connelly asked [9] whether the same result holds true if the genericity assumption of p is replaced by the weaker assumption of general position. We answer this question in the affirmative in this paper.