We study planar polygonal curves with the variational methods. We show a unified interpretation of discrete curvatures and the Steiner-type formula by extracting the notion of the discrete curvature vector from the first variation of the length functional. Moreover, we determine the equilibrium curves for the length functional under the area-constraint condition and study their stability.

中文翻译：

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关于平衡离散平面曲线的稳定性问题

我们用变分方法研究平面多边形曲线。我们通过从长度泛函的第一个变体中提取离散曲率向量的概念，展示了对离散曲率和 Steiner 型公式的统一解释。此外，我们确定了面积约束条件下长度泛函的平衡曲线并研究了它们的稳定性。