Robust Tensegrity Polygons

A tensegrity polygon is a planar cable-strut tensegrity framework in which the cables form a convex polygon containing all vertices. The underlying edge-labeled graph $$T=(V;C,S)$$T=(VźC,S), in which the cable edges form a Hamilton cycle, is an abstract tensegrity polygon. It is said to be robust if every convex realization of T as a tensegrity polygon has an equilibrium stress which is positive on the cables and negative on the struts, or equivalently, if every convex realization of T is infinitesimally rigid. We characterize the robust abstract tensegrity polygons on n vertices with $$n-2$$n-2 struts, answering a question of Roth and Whiteley (Trans Am Math Soc 265:419---446, 1981) and solving an open problem of Connelly (Recent progress in rigidity theory, 2008).