A Novel Two-Dimensional Distance Metric to Generalize Manhattan Distance

Abstract

Distance is a fundamental definition in fields such as geometry, mathematics, and physics. Because it is a very fundamental metric, it is not easy to create a new definition. In this study, we analyze existing distance definition and propose to generalize Euclidean distance and Manhattan distance, which are mainly used distance metric in existing distance definition. We analyze the definition of Minkowski distance, which is previously used as a concept of generalization along with Chebyshev distance, and the disadvantages of using this distance metric. By introducing a new perspective that interprets the existing Manhattan distance as a distance measured in four axes rather than simply adding the distances in each axis direction, this research introduces a new distance metric in two dimensions. This is a metric that generalizes the Euclidean distance and the Manhattan distance, and the proposed distance metric is derived from a geometrical aspect and an algorithm for calculating it is presented. We applied the existing distance definition and compared the differences through the results of generating a Voronoi area by the shortest distance from randomly distributed points in two dimensions. It is expected that the proposed method can be applied and expanded to the field of various graphics algorithms that use the distance metric.