Classification of Conic Sections in $PE_2(\mathbb{R})$

TL;DR

This paper offers a comprehensive and complete classification of conics in pseudo-Euclidean plane $PE_2( eal)$, introducing new invariants and methods for their analysis, which improves upon previous incomplete classifications.

Contribution

It provides a complete classification of conics in $PE_2( eal)$, including invariants and geometric families, enabling analysis without reducing to canonical forms.

Findings

Complete classification of conics in $PE_2( eal)$

Introduction of pseudo-orthogonal matrices and invariants

A comprehensive overview table of conic types

Abstract

This paper gives a complete classification of conics in $PE_2(\mathbb{R})$. The classification has been made earlier (Reveruk [5]), but it showed to be incomplete and not possible to cite and use in further studies of properties of conics, pencil of conics, and of quadratic forms in pseudo-Euclidean spaces. This paper provides that. A pseudo-orthogonal matrix, pseudo-Euclidean values of a matrix, diagonalization of a matrix in a pseudo-Euclidean way are introduced. Conics are divided in families and by types, giving both of them geometrical meaning. The invariants of a conic with respect to the group of motions in $PE_2(\mathbb{R})$ are determined, making it possible to determine a conic without reducing its equation to canonical form. An overview table is given.