Distances in Residential Space: Implications from Estimated Metric Functions for Minimum Path Distances

Abstract

To date, theories and conceptual frameworks formulated in geography and regional science assume Euclidean space frequently, Riemannian space (i.e., a landscape occurs on a sphere, and hence great circle distance measures separation) rarely, or networks embedded in Euclidean space, which is non-Euclidean in nature. Minkowskian space furnishes another family of non-Euclidean spaces, with both Euclidean and Manhattan space being special cases of it. This paper explores the non-Euclidean nature of network spaces, including Lobachevskian space (via the Poncaŕe disc). Metric functions for the various spaces are estimated for minimum path distances across various ideal and empirical networks, including selected regional and urban-region transport arteries, limited-access urban mass transit rail systems, and limited-access neighborhoods. The regional and urban-region landscapes involve nested transportation arteries, with findings addressing the question of whether or not increasing road density moves a network closer to Euclidean space. The general conclusion is that skewed transportation networks in large-scale regional landscapes may well be best characterized by a Lobachevskian geometry, whereas most smaller scale landscapes appear to be best characterized by a Minkowskian space whose parameters are between those of a Manhattan and a Euclidean space. The principal implication is that geography and regional science theories might furnish additional insights into the real world if the assumption of Euclidean space is replaced by one of Manhattan space.