Iterative Universal Rigidity

Robert Connelly, Steven J. Gortler

Abstract: A bar framework determined by a finite graph $$G$$ G and a configuration $$\mathbf{p =(p_1,\ldots , p_n) }$$ p = ( p 1 , … , p n ) in $$\mathbb {R}^d$$ R d is universally rigid if it is rigid in any $$\mathbb {R}^D \supset \mathbb {R}^d$$ R D ⊃ R d . We provide a characterization of universal rigidity for any graph $$G$$ G and any configuration $$\mathbf{p}$$ p in terms of a sequence of affine subsets of the space of configurations. This corresponds to a facial reduction process for closed finite-dimensional convex cones.