Contextual Metrics a Mathematical Definition for a Comprehensive Approach of Geographical Distances

Our goal is to establish a mathematical framework for the description of geographical distance in a comprehensive way. Geographical distance always refer to potential or realized movement between places, and these displacements obey the least effort rule. While this optimization of effort is well known to imply the Triangle Inequality in many situation, breaks in movement generate a paradox: effort optimization, taking into account the need to rest, results in apparent violations of the Triangle Inequality. In order to solve this issue, we introduce contextual metrics that consider space but also any contextual information relevant to travel, such as resources used for moving. Our approach permits to build a subjective space where distances are affected by the characteristics of the individual on the move. Contextual metrics frame the optimization problem in a space enriched by the context that the traveler has to take into account, making apparent that the violation of the Triangle Inequality in case of break was only an artifact of a model lacking crucial information. The range of geographical situations that can be modeled with this framework underline the level of generalization that can be expected from this approach.