Geometric Sensitivity of Rigid Graphs

Let $(G,p)$ be a minimally 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 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 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 $v$. (ii) Let $S$ be a designated set of joints and suppose that each joint with a nonzero load belongs to $S$. The active zone of $S$ is the set of those bars in which the stress, which resolves $L$, is nonzero. We prove that if $(G,p)$ is generic, then for almost all loads these zones depend only on the graph $G$ of the framework. These results are extended to arbitrary infinitesimally rigid generic frameworks. We also show that for $d=2$ these zones can be computed by efficient combinatorial methods.