Geometric Measures

A great variety of measures, many still in their early development, are used to describe the geometric properties of an object/distribution in geographic space. Some are derived directly from geometry, such as length, perimeter, and area, whereas others have evolved in the study of geography, for example, the compactness index. Unlike their geometric counterparts, objects/distributions in geographic space appear complex and diverse, depending on their point composition (e.g., cities), lines (railway networks), or areas (e.g., the territory of the Philippines). Furthermore, little agreement exists among spatial analysts on the classification of the various geometric measures. The history of these measures dates to the “Common Era,” when Strabo recognized the difficulty of measuring the shape of Italy. Five categories of geometric properties are recognized in the literature: (1) centroid, (2) orientation, (3) range, (4) intensity (or compactness), and (5) shape. Centroid is the center point of an object/distribution in geographic space, the point at which the object/distribution attains balance; orientation is the direction along which an object/distribution is arranged over space; range is the two-dimensional (2D) scope (spread) of an object; intensity is the extent of compactness (closeness) among elements of an object/distribution; and shape is what is left after the effects of position, orientation, and size are removed.

Various indices are used to measure each of these geometric properties. Centroid measures include the mean center and its extensions to global coordinate systems; orientation measures include the direction of the major axis of the standard distance ellipse; range measures include the mean distance, standard distance, standard distance deviation, and standard distance ellipse; intensity measures include compactness and spatial intensity indices; and shape measures include the elongation, circularity, overlap indices, ratio of the length of the major to the minor axis of the standard distance ellipse, and others. Notably, the standard distance ellipse can show simultaneously all the five geometric properties on a map except shape; and the centroid usually serves as a reference for calculating the other geometric measures.

Enormous difficulties have been encountered in examining the geometric properties of spatial objects, including the creation of a coherent framework. It has also been difficult to separate some properties from others; for example, measures of shape, compactness, and orientation are often mixed. Furthermore, geometric properties vary with scale; for example, the length of a river increases as scale decreases. Finally, the curvature of the globe creates problems; for example, the standard distance ellipse does not apply to spatial analysis in spherical space.

These geometric measures follow to varying extents the laws of addition and transitivity. Unlike length and area, each of which are additive across spatial objects, the five categories of geometric measures are not additive. But the rule of transitivity generally applies, except in the case of the shape indices. Transitivity, for instance, increases with the levels of similarity between two shapes.