What is it about?
The idea that geographical distances are often misunderstood in scientific literature is proved by an analysis of scientific --mainly geographic and economic-- literature. The key property of metrics for spatiality is triangle inequality (TI), which is closely related to the distance optimality. We identify three different situations where several authors identify violations of TI.We consider two of them to be errors of interpretation. The first error consists in considering sub-optimal measurements as distances. Yet distances are necessarily optimal if they obey TI. The second set of errors, which is the most widespread, entails a confusion between the Euclidean straight line and the shortest path. The errors lie in treating a detour as a violation of TI, whereas this situation simply corresponds to a non-Euclidean distance. The third problem concerns the additivity of distances. The commonplace situation in geographical space where a break is needed to provide the energy necessary to renew movement, is considered by some authors as another violation of TI. We argue that if these routes are optimal, TI should hold.
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Why is it important?
Distance is a central concept in geography and is a key concept in economics. Its correct understanding is hence a fundamental work for these disciplines.
Perspectives
This discussion provides several new and even somewhat counter-intuitive perspectives on three elements of spatiality and movement: the optimality of distances and the role of detours and breaks in contributing to this optimality. This work shows that most distances studied up to now have actually been metrics in the mathematical sense. Moreover, our investigation tends to indicate that geographical distances should be modelled as metrics or quasi-metrics, i.e. that triangle inequality should be observed in distance measurements. In the same vein, an unverified hypothesis that needs to be tested is that, for a distance to be validated as a geographical distance, it must be possible for the space in question to be projected coherently, or to put it more generally, to be transformed in the sense employed by (Tobler 1961; Ahmed and Miller 2007) in a form of Euclidean or non-Euclidean cartography, as set out in (L’Hostis 2009). According to this hypothesis, for a given distance to be treated as geographical distance, the underlying space should allow for consistent transformation of the set of geographical locations without topological disruption.
Dr Alain L'Hostis
Université Gustave Eiffel
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This page is a summary of: Misunderstanding geographical distances: two errors and an issue in the interpretation of violations of triangle inequality, Cybergeo, November 2016, OpenEdition,
DOI: 10.4000/cybergeo.27810.
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