Rad HAZU, Matematičke znanosti, Vol. 18 (2014), 125-143.

CLASSIFICATION OF CONIC SECTIONS IN PE2(R)

Jelena Beban-Brkić and Marija Šimić Horvath

Faculty of Geodesy, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: jbeban@geof.hr

Faculty of Architecture, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: marija.simic@arhitekt.hr

Abstract.   This paper gives a complete classification of conics in PE2(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 PE2(R) are determined, making it possible to determine a conic without reducing its equation to canonical form. An overview table is given.

2010 Mathematics Subject Classification.   51A05, 51N25.

Key words and phrases.   Pseudo-Euclidean plane PE2(R), conic section.

Full text (PDF) (free access)

References:

  1. H. Anton and C. Rorres, Elementary Linear Algebra: Applications Version, John Wiley and Sons, Inc., New York, 2000.

  2. J. Beban-Brkić, Isometric invariants of conics in the isotropic plane-classification of conics, J. Geom. Graph. 6 (2002), 17-26.
    MathSciNet

  3. G. S. Birman and K. Nomizu, Trigonometry in Lorentzian Geometry, Amer. Math. Monthly 91 (1984), 543-549.
    MathSciNet     CrossRef

  4. I. M. Yaglom, Princip otnositeljnosti Galileja i neevklidova geometrija, Nauka, Moskva, 1969.

  5. N. V. Reveruk, Krivie vtorogo porjadka v psevdoevklidovoi geometrii, Uchenye zapiski MPI 253 (1969), 160-177.
    MathSciNet

  6. V. G. Shervatov, Hyperbolic functions, Dover Publications, New York, 2007.

Rad HAZU Home Page