A Constrained Optimization Approach for Constructing Rigid Bar Frameworks with Higher-order Rigidity
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| author | Li, Xuenan Santangelo, Christian D. Holmes-Cerfon, Miranda |
| author_facet | Li, Xuenan Santangelo, Christian D. Holmes-Cerfon, Miranda |
| contents | We present a systematic approach for constructing bar frameworks that are rigid but not first-order rigid, using constrained optimization. We show that prestress stable (but not first-order rigid) frameworks arise as the solution to a simple optimization problem, which asks to maximize or minimize the length of one edge while keep the other edge lengths fixed. By starting with a random first-order rigid framework, we can thus design a wide variety of prestress stable frameworks, which, unlike many examples known in the literature, have no special symmetries. We then show how to incorporate a bifurcation method to design frameworks that are third-order rigid. Our results highlight connections between concepts in rigidity theory and constrained optimization, offering new insights into the construction and analysis of bar frameworks with higher-order rigidity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23072 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Constrained Optimization Approach for Constructing Rigid Bar Frameworks with Higher-order Rigidity Li, Xuenan Santangelo, Christian D. Holmes-Cerfon, Miranda Metric Geometry Optimization and Control We present a systematic approach for constructing bar frameworks that are rigid but not first-order rigid, using constrained optimization. We show that prestress stable (but not first-order rigid) frameworks arise as the solution to a simple optimization problem, which asks to maximize or minimize the length of one edge while keep the other edge lengths fixed. By starting with a random first-order rigid framework, we can thus design a wide variety of prestress stable frameworks, which, unlike many examples known in the literature, have no special symmetries. We then show how to incorporate a bifurcation method to design frameworks that are third-order rigid. Our results highlight connections between concepts in rigidity theory and constrained optimization, offering new insights into the construction and analysis of bar frameworks with higher-order rigidity. |
| title | A Constrained Optimization Approach for Constructing Rigid Bar Frameworks with Higher-order Rigidity |
| topic | Metric Geometry Optimization and Control |
| url | https://arxiv.org/abs/2509.23072 |