Geometric Sensitivity of Rigid Graphs

Let (G, p) be an infinitesimally rigid d-dimensional bar-and-joint framework and let L be an equilibrium load on p. The load can be resolved by appropriate stresses wi,j , ij ∈ E(G), in the bars of the framework. Our goal is to identify the following parts (zones) of the framework: (i) when the location of an unloaded joint v is slightly perturbed, and the same load is applied, the stress will change in some of the bars. We call the set of these bars the influenced zone of v (with respect to L, p and the modified configuration p′), (ii) let S be a designated set of joints and suppose that each joint with a non-zero load belongs to S. The active zone of S (with respect to p and L) is the set of those bars in which the stress, which resolves L, is non-zero. We show that if (G, p) is generic and d = 2 then, for almost all loads, these zones depend only on the graph G of the framework and can be computed by efficient combinatorial methods.