Measuring the form and structure of geographic landscapes is fundamental to understanding geospatial phenomena and their dynamics. Shape compactness metrics have been extensively employed in gerrymandering assessment, urban planning, landscape ecology, and other applications. Existing compactness measurements for gerrymandering, however, target particular aspects of gerrymandered shapes such as elongation, indentation, or dispersion without adequately integrating them with spatial context. This article proposes a comprehensive shape compactness metric that coherently integrates these aspects. It first divides a district into nonoverlapping maximum inscribed circles. Then Euclidean distances from the centers of these circles to the district centroid are standardized and regulated using contextual and topological factors like the distribution of attributes, the fixed upper level boundary, and the relationships with other districts. Finally, these regulated distances are aggregated to produce a quantitative compactness measure that, unlike most existing ones, features a threshold for gerrymandering identification and the coherent integration of roundness, convexity, closeness, and spatial context. Applying the new metric to the U.S. Congressional districts exemplifies its differences from existing metrics and also illustrates the necessity and value of coherently combining multiple aspects of gerrymandering in a single shape compactness metric.

测量地理景观的形式和结构是了解地理空间现象及其动态的基础。形状紧凑度度量标准已广泛用于搬运评估，城市规划，景观生态学和其他应用中。然而，现有的用于压纹的压实度测量针对的是压纹形状的特定方面，例如伸长率，压痕或色散，而没有将它们与空间环境充分融合。本文提出了一个综合的形状紧凑性度量标准，将这些方面紧密结合在一起。它首先将一个区域划分为不重叠的最大内切圆。然后使用上下文和拓扑因素（例如属性的分布，固定的高层边界以及与其他地区的关系）来标准化和调节从这些圆的中心到地区质心的欧几里得距离。最后，将这些调节距离进行汇总，以产生定量紧凑性度量，该度量与大多数现有度量不同，其特征是用于进行识别识别的阈值以及圆度，凸度，亲密性和空间上下文的连贯集成。将新度量标准应用于美国国会区，可以说明其与现有度量标准的差异，还说明了在单一形状紧凑性度量标准中将皂酒的多个方面连贯地结合起来的必要性和价值。以及与其他地区的关系。最后，将这些调节距离进行汇总，以产生定量紧凑性度量，该度量与大多数现有度量不同，其特征是用于进行识别识别的阈值以及圆度，凸度，亲密性和空间上下文的连贯集成。将新度量标准应用于美国国会区，可以说明其与现有度量标准的差异，还说明了在单一形状紧凑性度量标准中将皂酒的多个方面连贯地结合起来的必要性和价值。以及与其他地区的关系。最后，将这些调节距离进行汇总，以产生定量紧凑性度量，该度量与大多数现有度量不同，其特征是用于进行识别识别的阈值以及圆度，凸度，亲密性和空间上下文的连贯集成。将新度量标准应用于美国国会区，可以说明其与现有度量标准的差异，还说明了在单一形状紧凑性度量标准中将皂酒的多个方面连贯地结合起来的必要性和价值。