engleski

On Classification of Conic Sections in the Pseudo- Euclidean Plane

A pseudo-Euclidean plane $PE_2(\mathbb{; ; R}; ; )$ is a real affine plane where a metric is induced by an absolute figure $(\omega, \Omega_1, \Omega_2)$ consisting of the line $\omega$ at infinity and two different real points $\Omega_1, \Omega_2 \in \omega$. The aim of our work is a complete classification of the second order curves in $PE_2(\mathbb{; ; R}; ; )$. The classification has been made earlier in the paper of N. V. Reveruk (Krivie vtorogo porjadka v psevdoevklidovoi geometrii, Uchenye zapiski MPI 253(1969) 160--177.), 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. Our approach is based on linear algebra. Notions such as a pseudo- orthogonal matrix, pseudo- Euclidean values of matrix, diagonalization of a matrix in a pseudo-Euclidean way are introduced. In addition, conics are divided in families and by types, giving both of them geometrical meaning. All this allows to determine the invariants of a conic with respect to the group of motions in $PE_2(\mathbb{; ; R}; ; )$ making it possible to determine a conic without reducing its equation to canonical form.

pseudo-Euclidean plane PE_2(R) ; conic section

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