Geometry of the conics on the Minkowski plane

TL;DR

This paper explores the geometry of conics in Minkowski spacetime, revealing how their properties can be understood through Euclidean perspectives and introducing a concept of extended geometric completeness.

Contribution

It introduces the notion of extended geometric completeness for conics in Minkowski space and demonstrates how conic completeness can be transformed via a Euclidean mirror.

Findings

Conics in Minkowski space can be interpreted using Euclidean geometry.

Extended geometric completeness provides new insights into conic properties.

Conic completeness can be altered through Euclidean transformations.

Abstract

Conics in the Euclidean space have been known for their geometrical beauty and also for their power to model several phenomena in real life. It usually happens that when thinking about the conics in a semi-Riemannian manifold, the equations and the graphs that come to mind are those of the quadratic Euclidean equations. For example, a circle is always perceived like a closed curve. We study the geometry of the conics in the semi-Riemannian Minkowski spacetime, and interpret each equation with Euclidean eyes. By defining an extended geometric completeness for conics, we will show that the conic completeness of conics can be changed through a Euclidean mirror.