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Geiringer–Laman Theorem
The Geiringer–Laman theorem gives a Structural rigidity, combinatorial characterization of Structural rigidity#Definitions, generically rigid graphs in 2-dimensional Euclidean space, with respect to Geometric constraint system#Bar-joint systems, bar-joint frameworks. This theorem was first proved by Hilda Geiringer, Hilda Pollaczek-Geiringer in 1927, and later by Gerard Laman in 1970. An efficient algorithm called the Pebble game (rigidity), pebble game is used to identify this class of graphs. This theorem has been the inspiration for many Geiringer-Laman type results for Structural rigidity#Rigidity for other types of frameworks, other types of frameworks with generalized pebble games. Statement of the theorem This theorem relies on definitions of genericity that can be found on the Structural rigidity#Definitions, structural rigidity page. Let V(E) denote the vertex set of a set of edges E. Geiringer-Laman Theorem. A graph G=(V,E) is Structural rigidity#Definitions, ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structural Rigidity
In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges. Definitions Rigidity is the property of a structure that it does not bend or flex under an applied force. The opposite of rigidity is flexibility. In structural rigidity theory, structures are formed by collections of objects that are themselves rigid bodies, often assumed to take simple geometric forms such as straight rods (line segments), with pairs of objects connected by flexible hinges. A structure is rigid if it cannot flex; that is, if there is no continuous motion of the structure that preserves the shape of its rigid components and the pattern of their connections at the hinges. There are two essentially different kinds of rigidity. Finite or macroscopic rigidity means that the structure will not flex, fold, or bend by a positive amount. Infinitesimal rigidity means that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
