A New Structural Rigidity for Geometric Constraint Systems

The structural rigidity property, a generalisation of Laman’s theorem which characterises generically rigid bar frameworks in 2D, is generally considered a good heuristic to detect rigidities in geometric constraint satisfaction problems (GCSPs). In fact, the gap between rigidity and structural rigidity is significant and essentially resides in the fact that structural rigidity does not take geometric properties into account. In this article, we propose a thorough analysis of this gap. This results in a new characterisation of rigidity, the extended structural rigidity, based on a new geometric concept: the degree of rigidity (DOR). We present an algorithm for computing the DOR of a GCSP, and we prove some properties linked to this geometric concept. We also show that the extended structural rigidity is strictly superior to the structural rigidity and can thus be used advantageously in the algorithms designed to tackle the major issues related to rigidity.

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