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Abstract
A discrete tensegrity framework can be thought of as a graph in Euclidean n-spacewhere each edge is of one of three types: an edge with a fixed length (bar) or an edgewith an upper (cable) or lower (strut) bound on its length. Roth and Whiteley, intheir 1981 paper Tensegrity Frameworks, showed that in certain cases, the strutsand cables can be replaced with bars when analyzing the framework for infinitesimalrigidity. In that case we call the tensegrity bar equivalent. In specific, they showedthat if there exists a set of positive weights, called a positive stress, on the edges suchthat the weighted sum of the edge vectors is zero at every vertex, then the tensegrityis bar equivalent.In this paper we consider an extended version of the tensegrity framework inwhich the vertex set is a (possibly infinite) set of points in Euclidean n-space and theedgeset is a compact set of unordered pairs of vertices. These are called continuoustensegrities. We show that if a continuous tensegrity has a strictly positive stress, itis bar equivalent and that it has a semipositive stress if and only if it is partially barequivalent. We also show that if a tensegrity is minimally bar equivalent (it is barequivalent but removing any open set of edges makes it no longer so), then it has astrictly positive stress.In particular, we examine the case where the vertices form a rectifiable curve andthe possible motions of the curve are limited to local isometries of it. Our methodsprovide an attractive proof of the following result: There is no locally arclengthpreserving motion of a circle that increases any antipodal distance without decreasingsome other one.