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The Algebraic Geometry of Motions of Bar-and-Body Frameworks

1987; Society for Industrial and Applied Mathematics; Volume: 8; Issue: 1 Linguagem: Inglês

10.1137/0608001

2168-345X

Neil White, Walter Whiteley,

Cellular Mechanics and Interactions

This paper generalizes and extends previous results on bar-and-joint frameworks to bar-and-body frameworks: structures formed by rigid bodies in space linked by rigid bars and universal joints. For a multi-graph which can form an isostatic (minimal infinitesimally rigid) bar-and-body framework, a single polynomial—the pure condition—is found which describes those bad positions of the bars for which infinitesimal rigidity fails. (The proof is much shorter than the previous derivation for bar-and-joint frameworks and the condition is linear in the variables.) The pure condition is used to describe the infinitesimal motions of a 1-underbraced framework in terms of the screw centers of motion of the bodies. The factoring of the polynomial condition is given by the lattice of isostatic blocks in the framework, with at most one irreducible factor for each block. For frameworks realized at generic points of an irreducible factor the infinitesimal motions and the static stresses are also given by the factoring and the lattice. (These results are much sharper than the corresponding results for bar-and-joint frameworks.) The theorems are presented in terms of k-frames—a simple generalization of bar-and-body frameworks which also has applications to scene analysis and other types of frameworks.