The purpose of this article is to examine how to measure the degree of similarity among various shapes, including for the first time those that are fragmented and perforated, by the overlap-based elongation index. It is argued that complete removal of the effects of position, size, and orientation on shape, which is essential for the calibration of shape similarity, can be achieved by a shape similarity index that varies continuously with changes in shape. After examining the characteristics of shape change, it is demonstrated that the elongation index is sensitive to changes in the shape of two spatial objects only when the centroids of the two objects are coincident. Two related rules of shape similarity are then presented. The applicability of the elongation index is evaluated by comparing several simple and complex shapes. The principal contribution of this article is that for the first time similarity among various shapes, fragmented or perforated, can be identified using the elongation index.

No abstract

This note presents a new measure of spatial imbalance that uses the center of gravity

This note presents a new measure of spatial imbalance that uses the center of gravity

Industrial agglomeration has attracted extensive attention from economists and geographers, yet it is still a challenge to identify the multi-agglomeration spatial structure and degree of industrial agglomeration in continuous space—there is still a lack of a more targeted industrial clustering method. The clustering method and the standard deviational ellipse (simply, ellipse) model have advantages in identifying the spatial structure and representing spatial information respectively. On this basis, we propose an ellipse-based approach to identifying industrial clusters. Our ellipse-based approach rests upon group nearest neighbor using the group-based nearest neighbor (GNN) ordering and spatial compactness matrix, where a number of point sequences with varying lengths, generated under the GNN ordering, are characterized by an ellipse and the elliptical parameters of these point sequences formulate the values and structure of the compactness matrix. Clustering is reformulated to identify ellipses with a specified parameter among a number of potential candidate ellipses, with significant changes (especially in the area) used as the cutoff criterion for determining the clusters’ border point. Our approach is illustrated in the location pattern of firms in Shanghai City, China in comparison with four well-known clustering methods. With the combination of elliptical parameters and spatial compactness, our approach may bring a new analytical ground for future industrial clustering research.