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On Classification of Conic Sections in the Pseudo- Euclidean Plane


Šimić Horvath, Marija; Beban-Brkić, Jelena
On Classification of Conic Sections in the Pseudo- Euclidean Plane // Book of Abstracts, 5th Croatian Mathematical Congress / Crnković, Dean ; Mikulić Crnković Vedrana ; Rukavina, Sanja (ur.).
Rijeka: University of Rijeka, 2012. str. 45-45 (predavanje, domaća recenzija, sažetak, znanstveni)


Naslov
On Classification of Conic Sections in the Pseudo- Euclidean Plane

Autori
Šimić Horvath, Marija ; Beban-Brkić, Jelena

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Book of Abstracts, 5th Croatian Mathematical Congress / Crnković, Dean ; Mikulić Crnković Vedrana ; Rukavina, Sanja - Rijeka : University of Rijeka, 2012, 45-45

ISBN
978-953-7720-13-1

Skup
5th Croatian Mathematical Congress

Mjesto i datum
Rijeka, Hrvatska, 18-21.06.2012

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Domaća recenzija

Ključne riječi
Pseudo-Euclidean plane PE_2(R) ; conic section

Sažetak
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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika



POVEZANOST RADA


Ustanove
Geodetski fakultet, Zagreb,
Arhitektonski fakultet, Zagreb