A fractal approach to identifying urban boundaries

Fractal geometry can be used for determining the morphological boundaries of metropolitan areas. A two-step method is proposed here: (1) Minkowski's dilation is applied to detect any multiscale spatial discontinuity and (2) a distance threshold is located on the dilation curve corresponding to a major change in its behavior. We therefore measure the maximum curvature of the dilation curve. The method is tested on theoretical urban patterns and on several European cities to identify their morphological boundaries and to track boundary changes over space and time. Results obtained show that cities characterized by comparable global densities may exhibit different distance thresholds. The less the distance separating buildings differ between an urban agglomeration and its surrounding built landscape, the greater the distance threshold. The few the buildings that are connected across scales, the greater the distance threshold.