Rigidity of frameworks
Yazarlar (1)
| Makale Türü | Özgün Makale (Uluslararası alan indekslerindeki dergilerde yayınlanan tam makale) | ||
| Dergi Adı | |||
| Makale Dili | – | Basım Tarihi | 03-2018 |
| Makale Linki | https://qmro.qmul.ac.uk/jspui/handle/123456789/36220 | ||
| UAK Araştırma Alanları |
Cebir ve Sayılar Teorisi
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| Özet |
| A d-dimensional (bar-and-joint) framework is a pair (G; p) where G = (V;E) is a graph and p : V > Rd is a function which is called the realisation of the framework (G; p). A motion of a framework (G; p) is a continuous function P : [0; 1] x V > Rd which preserves the edge lengths for all t 2 [0; 1]. A motion is rigid if it also preserves the distances between non-adjacent pairs of vertices of G. A framework is rigid if all of its motions are rigid motions. An in nitesimal motion of a d-dimensional framework (G; p) is a function q : V > Rd such that [p(u) - p(v)] ~ [q(u) - q(v)] = 0 for all uv 2 E. An in nitesimal motion of the framework (G; p) is rigid if we have [p(u) - p(v)] . [q(u) - q(v)] = 0 also for non-adjacent pairs of vertices. A framework (G; p) is in nitesimally rigid if all of its in nitesimal motions are rigid in nitesimal motions. A d-dimensional framework (G; p) is generic if the coordinates of the positions of vertices assigned by p are algebraically independent. For generic frameworks rigidity and in nitesimal rigidity are equivalent. We construct a matrix of size |E| xd|V| for a given d-dimensional framework (G; p) as follows. The rows are indexed by the edges of G and the set of d consecutive columns corresponds to a vertex of G. The entries of a row indexed by uv 2 E contain the d coordinates of p(u) - p(v) and p(v) - p(u) in the d consecutive columns corresponding to u and v, respectively, and the remaining entries are all zeros. This matrix is the rigidity matrix of the framework (G; p) and denoted by R(G; p). Translations and rotations of a given framework (G; p) give rise to a subspace of dimension d+1 2 of the null space of R(G; p) when p(v) affinely spans Rd. Therefore we have … |
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