Misunderstanding geographical distances: two errors and an issue in the interpretation of violations of triangle inequality

Abstract

We investigate the meaning of the mathematical properties of distances in the fields of geography and economics. 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.

Keywords

• distance
• mathematics
• geography
• spatial economy
• transportation