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For rigid geometry, see Rigid analytic space. For other notions of rigidity in geometry, see Mostow rigidity theorem and Rigidity (mathematics).
In discrete geometry, geometric rigidity is a theory for determining if a geometric constraint system (GCS) has finitely many -dimensional solutions, or frameworks, in some metric space. A framework of a GCS is rigid in -dimensions, for a given if it is an isolated solution of the GCS, factoring out the set of trivial motions, or isometric group, of the metric space, e.g. translations and rotations in Euclidean space. In other words, a rigid framework of a GCS has no nearby framework of the GCS that is reachable via a non-trivial continuous motion of that preserves the constraints of the GCS. Structural rigidity is another theory of rigidity that concerns generic frameworks, i.e., frameworks whose rigidity properties are representative of all frameworks with the same constraint graph. Results in geometric rigidity apply to all frameworks; in particular, to non-generic frameworks.
Left: a generically rigid graph in . Assigning the edge the distance results in a family of non-generic flexible bar-joint systems. Right: a flexible framework of such a system.
Geometric rigidity was first explored by Euler, who conjectured that all polyhedra in -dimensions are rigid. Much work has gone into proving the conjecture, leading to many interesting results discussed below. However, a counterexample was eventually found. There are also some generic rigidity results with no combinatorial components, so they are related to both geometric and structural rigidity.
and 23 Related for: Geometric rigidity information
In discrete geometry, geometricrigidity is a theory for determining if a geometric constraint system (GCS) has finitely many d {\displaystyle d} -dimensional...
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working on other rigidity phenomena, and in collaboration with several colleagues, made contributions to smooth rigidity and geometricrigidity, to differential...
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Margulis, who established their global rigidity results out of attempts to understand infinitesimal rigidity results such as Calabi and Vesentini's,...
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supports. Like the truss, a space frame is strong because of the inherent rigidity of the triangle; flexing loads (bending moments) are transmitted as tension...
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already been discovered in 1927 by Hilda Geiringer. Laman graphs arise in rigidity theory: if one places the vertices of a Laman graph in the Euclidean plane...
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