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

### CLASSIFICATION OF CONIC SECTIONS IN PE_{2}(**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
*PE*_{2}(**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}(**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 *PE*_{2}(**R**), conic section.

**Full text (PDF)** (free access)

**References:**

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

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

MathSciNet

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

MathSciNet
CrossRef

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

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

MathSciNet

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

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