A Tractable, Approximate, Combinatorial 3D rigidity characterization

There is no known, tractable, characterization of 3D rigidity of sets of points constrained by pairwise distances or 3D distance constraint graphs. We give a combinatorial approximate characterization of such graphs which we call module-rigidity, which can be determined by a polynomial time algorithm. We show that this property is natural and robust in a formal sense. Rigidity implies module-rigidity, and modulerigidity significantly improves upon the generalized Laman degree-of-freedom or density count. Specifically, graphs containing ”bananas” or ”hinges” [8] are not module-rigid, while the generalized Laman count would claim rigidity. The algorithm that follows from our characterization of module-rigidity gives a complete decomposition of non module-rigid graphs into its maximal module-rigid subgraphs. To put the result in perspective, it should be noted that, prior to the recent algorithm of [21] there was no known polynomial time algorithm for obtaining all maximal subgraphs of an input constraint graph that satisfy the generalized Laman count, specifically when overconstraints or redundant constraints are present. The new method has been implemented in the FRONTIER [23], [35], [28], [29] opensource 3D geometric constraint solver and has many useful properties and practical applications [30], [31], [32], [34], [33]. Specifically, the method is used for constructing a so-called decomposition-recombination (DR) plan for 3D geometric constraint systems, which is crucial to defeat the exponential complexity of solving the (sparse) polynomial system obtained from the entire geometric constraint system. The DR-plan guides the algebraic-numeric solver by ensuring that only small subsystems are ever solved. The new, approximate characterization of 3D rigidity permits FRONTIER to deal with a far larger class of 3D constraint systems (a class adequate for most applications) than any other current geometric constraint solver.