Integrating overlay and spatial optimization in assessing shape compactness

GIScience has been central in the evaluation and assessment of geographic shape. Reasons for this are many, but no doubt this is driven by the ease of mapping and visualization of an area that enables human reasoning about its shape characteristics. Coupled with the recognition that the most compact shape is a circle, this has led to the development of much theory and associated methods for quantifying the compactness of an area or polygon. An important class of approaches for assessing compactness involves the comparison of a polygon to a corresponding circle. However, for many approaches, left unresolved is what the appropriate circle of comparison is, and in particular, how best to position and size the circle in order to meaningfully quantify shape compactness. This paper formalizes the sizing and positioning problem essential for the assessment of compactness. An approach is developed to structure and solve this using spatial optimization. Empirical assessment highlights the importance and significance of correctly sizing and positioning a comparative circle. Additionally, it is shown that this approach can be readily implemented and applied in any commercial or open-source GIS environment.