Minkowski Metrics: Its Interaction and Complementarity with Euclidean Metrics

Abstract

A certain set of objects has a Minkowski space structure when these objects can be represented by coordinates (x ₀, x ₁, …, x _n ) with $({x}_{1}^{2} + \cdots + {x}_{n}^{2}) - {x}_{0}^{2} \leq 0$. This chapter considers three situations where this structure and its associated tools bring a new light. It first introduces a well-known such space, having its roots in Minkowski’s work, the space–time representation. Then, it considers a 3D representation space, S ₂ ⁺, for 2 ×2 symmetrical positive definite matrices, the determinant of these matrices providing this space with a Minkowski structure. An application to a representation of discrete curves is detailed. The last example is another 3D representation space, denoted Ω ₂, for plane circles in the Euclidean space ℝ ². The natural Minkowski quadratic form for a circle in Ω ₂ is the square of its radius, for a convenient set of coordinates. An application is given to a new measure of circularity for a finite set of points.