Infinitesimal rigidity and prestress stability for frameworks in normed spaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866913198287880192 |
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| author | Dewar, Sean |
| author_facet | Dewar, Sean |
| contents | A (bar-and-joint) framework is a set of points in a normed space with a set of fixed distance constraints between them. Determining whether a framework is locally rigid - i.e. whether every other suitably close framework with the same distance constraints is an isometric copy - is NP-hard when the normed space has dimension 2 or greater. We can reduce the complexity by instead considering derivatives of the constraints, which linearises the problem. By applying methods from non-smooth analysis, we shall strengthen previous sufficient conditions for framework rigidity that utilise first-order derivatives. We shall also introduce the notions of prestress stability and second-order rigidity to the topic of normed space rigidity, two weaker sufficient conditions for framework rigidity previously only considered for Euclidean spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_14468 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Infinitesimal rigidity and prestress stability for frameworks in normed spaces Dewar, Sean Metric Geometry 52C25 (Primary) 52A21, 49J52 (Secondary) A (bar-and-joint) framework is a set of points in a normed space with a set of fixed distance constraints between them. Determining whether a framework is locally rigid - i.e. whether every other suitably close framework with the same distance constraints is an isometric copy - is NP-hard when the normed space has dimension 2 or greater. We can reduce the complexity by instead considering derivatives of the constraints, which linearises the problem. By applying methods from non-smooth analysis, we shall strengthen previous sufficient conditions for framework rigidity that utilise first-order derivatives. We shall also introduce the notions of prestress stability and second-order rigidity to the topic of normed space rigidity, two weaker sufficient conditions for framework rigidity previously only considered for Euclidean spaces. |
| title | Infinitesimal rigidity and prestress stability for frameworks in normed spaces |
| topic | Metric Geometry 52C25 (Primary) 52A21, 49J52 (Secondary) |
| url | https://arxiv.org/abs/2109.14468 |