A well-known combinatorial algorithm can decide generic rigidity in the plane by determining if the graph is of Pollaczek–Geiringer–Laman type. Methods from matroid theory have been used to prove other interesting results, again under the assumption of generic configurations. However, configurations arising in applications may not be generic. We present Theorem 4.2 and its corresponding Algorithm 1 which decide if a configuration is 𝜀-locally rigid, a notion we define. A configuration which is 𝜀-locally rigid may be locally rigid or flexible, but any continuous deformations remain within a sphere of radius 𝜀 in configuration space. Deciding 𝜀-local rigidity is possible for configurations which are smooth or singular, generic or non-generic. We also present Algorithms 2 and 3 which use numerical algebraic geometry to compute a discrete-time sample of a continuous flex, providing useful visual information for the scientist.

中文翻译：

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Epsilon 局部刚度和数值代数几何

一种著名的组合算法可以通过确定图形是否属于 Pollaczek-Geiringer-Laman 类型来确定平面中的一般刚度。拟阵理论的方法已被用于证明其他有趣的结果，同样是在通用配置的假设下。但是，应用程序中出现的配置可能不是通用的。我们提出了定理 4.2 及其相应的算法 1，它们决定了一个配置是否是𝜀-局部刚性，我们定义的一个概念。一个配置是𝜀- 局部刚性可以是局部刚性或柔性的，但任何连续变形都保持在半径范围内𝜀在配置空间。决定𝜀-局部刚性对于平滑或奇异、通用或非通用的配置是可能的。我们还介绍了算法 2 和 3，它们使用数值代数几何来计算连续弯曲的离散时间样本，为科学家提供有用的视觉信息。