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arXiv · 2509.23072

A Constrained Optimization Approach for Constructing Rigid Bar Frameworks with Higher-order Rigidity

Xuenan Li·Christian D. Santangelo·Miranda Holmes-Cerfon
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Abstract

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.

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Xuenan Li, Christian D. Santangelo, Miranda Holmes-Cerfon. 2025-09-27. A Constrained Optimization Approach for Constructing Rigid Bar Frameworks with Higher-order Rigidity. https://arxiv.org/abs/2509.23072

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