Recently proposed as a stable means of evaluating geometric compactness, the isoperimetric profile of a planar domain measures the minimum perimeter needed to inscribe a shape with prescribed area varying from 0 to the area of the domain. While this profile has proven valuable for evaluating properties of geographic partitions, existing algorithms for its computation rely on aggressive approximations and are still computationally expensive. In this paper, we propose a practical means of approximating the isoperimetric profile and show that for domains satisfying a "thick neck" condition, our approximation is exact. For more general domains, we show that our bound is still exact within a conservative regime and is otherwise an upper bound. Our method is based on a traversal of the medial axis which produces efficient and robust results. We compare our technique with the state-of-the-art approximation to the isoperimetric profile on a variety of domains and show significantly tighter bounds than were previously achievable.

中文翻译：

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中轴等周轮廓

最近提出作为评估几何紧凑性的稳定方法，平面域的等周轮廓测量内切形状所需的最小周长，指定面积从 0 到域的面积变化。虽然此配置文件已被证明对评估地理分区的属性很有价值，但用于其计算的现有算法依赖于积极的近似，并且计算成本仍然很高。在本文中，我们提出了一种逼近等周剖面的实用方法，并表明对于满足“粗颈”条件的域，我们的逼近是准确的。对于更一般的领域，我们表明我们的界限在保守体制内仍然是精确的，否则是上限。我们的方法基于中轴的遍历，从而产生高效且稳健的结果。