Geometric rigidity

In discrete geometry, geometric rigidity is a theory for determining if a geometric constraint system (GCS) has finitely many d {\displaystyle d} -dimensional solutions, or frameworks, in some metric space. A framework of a GCS is rigid in d {\displaystyle d} -dimensions, for a given d {\displaystyle d} 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 ( G , p ) {\displaystyle (G,p)} of a GCS has no nearby framework of the GCS that is reachable via a non-trivial continuous motion of ( G , p ) {\displaystyle (G,p)} 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.

Geometric rigidity was first explored by Euler, who conjectured that all polyhedra in 3 {\displaystyle 3} -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.

Definitions The definitions below, which can be found in, are with respect to bar-joint frameworks in d {\displaystyle d} -dimensional Euclidean space, and will be generalized for other frameworks and metric spaces as needed. Consider a linkage ( G , δ ) {\displaystyle (G,\delta )} , i.e. a constraint graph G = ( V , E ) {\displaystyle G=(V,E)} with distance constraints δ {\displaystyle \delta } assigned to its edges, and the configuration space C ( G , δ ) {\displaystyle {\mathcal {C}}(G,\delta )} consisting of frameworks ( G , p ) {\displaystyle (G,p)} of ( G , δ ) {\displaystyle (G,\delta )} . The frameworks in C ( G , δ ) {\displaystyle {\mathcal {C}}(G,\delta )} consist of maps p : V → R d | V | {\displaystyle p:V\rightarrow \mathbb {R} ^{d|V|}} that satisfy

‖ p ( u ) − p ( v ) ‖ 2 = δ u v , {\displaystyle \|p(u)-p(v)\|^{2}=\delta _{uv},}

… excerpt ends here. Continue reading the full article.