On the Fractal Measurement of Geographical Boundaries

This paper has two main objectives. First, we review and evaluate four different computational methods for measuring the “fractality” of cartographic lines: these are known as the structured walk, the equipaced polygon, the hybrid walk, and the cell count methods. Second, because previous research has restricted the focus of fractal measurement exercises to isolated physical entities, the subject of our empirical study concerns the urban area of Swindon, United Kingdom, that comprises a mosaic of contiguous land-use parcels. In technical terms, the results pinpoint some of the comparative strengths and weaknesses of the four methods, whilst our substantive conclusion is that fractal dimension appears to be a function of both scale and land-use type in our geographical study.     

Minimum-Cost Distances in Spatial Analysis

An axiomatic approach to distance is developed which focuses on those behavioral concepts of distance related to movement in space. In particular, spatial movement by behaving units is postulated to involve a choice from among some set of abstract trips in space, and implicitly, to involve the minimization of some relevant notion of trip costs. In this context, the relevant behavioral notion of distance in space is taken to be the minimum-cost distance generated by this choice process. These trip-cost concepts extend the classical notions of paths, path lengths, and shortest paths in metric spaces. Hence many of the analytical results of the paper involve extensions of classical shortest-path distance properties to minimum-cost distances. In addition to these extensions, a characterization theorem is given which specifies the possible functional relationships between trip costs and their associated path lengths. These relationships include most functional forms which are commonly employed in the literature.