Rigidity Theories

A framework in Ed consists of a configuration of vertices some pair of which are joined by rigid "rods." A framework is (infinitesimally) rigid if there are no small perturbations that preserve the lengths of all the rods (other than congruences of the whole structure). Figure 1 shows some examples for rigid and non-rigid frameworks.

For a framework of e rods and v vertices where the rods and vertices of the framework are the edges and vertices of a convex polyhedron, such polyhedron has (2 + e - v) faces by Euler's relationship (click here for proof.) Then, it follows from Cauchy's rigidity theorem (see below) that if the faces of the above polyhedron are all triangular, i.e., if 3v - 6 = e, then the convex polyhedral structure is rigid and not suspectible to small perturbations.

Cauchy's Rigidity Theorem

Cauchy proved that any convex polyhedron in R3 with rigid faces, but hinged at the edges, was in fact completely rigid (i.e., it could not be subjected to small perturbations).(click here for proof.)

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Illustrated by Jonathan Shum
Center for Intelligent Machines
McGill University, Montreal, Canada.

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