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.

中文翻译：

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上下文度量是地理距离综合方法的数学定义

我们的目标是建立一个全面描述地理距离的数学框架。地理距离总是指地方之间潜在的或已实现的移动，这些位移遵循最小努力规则。虽然众所周知，这种努力的优化在许多情况下暗示了三角不等式，但运动中断会产生一个悖论：努力优化，考虑到休息的需要，导致明显违反三角不等式。为了解决这个问题，我们引入了考虑空间以及与旅行相关的任何上下文信息的上下文度量，例如用于移动的资源。我们的方法允许建立一个主观空间其中距离受移动中个人特征的影响。上下文度量在一个空间中构建优化问题，该空间由旅行者必须考虑的上下文丰富，这表明在中断情况下违反三角不等式只是缺乏关键信息的模型的产物。可以用这个框架建模的地理情况范围强调了可以从这种方法中预期的概括水平。