On affine motions and bar frameworks in general position

A configuration p in r-dimensional Euclidean space is a finite collection of points (p(1), ... , p(n)) that affinely span R-r. A bar framework. denoted by G(p), in R-r is a simple graph G on n vertices together with a configuration p in R-r. 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 vertical bar vertical bar q(i) - q(j)vertical bar vertical bar = vertical bar vertical bar p(i) - p(j)vertical bar vertical bar for each edge (i, j) of G. It is known [2,7] that if configuration p is generic and bar framework G(p) in R-r 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. (C) 2012 Elsevier Inc. All rights reserved.