engleski

Isometric Invariants of Conics in Isotropic Plane - Classification of Conics

A real affine plane A_2 is called an isotropic plane I_2, if in A_2 a metric is induced by an absolute {;f, F};, consisting of the line at infinity f of A_2 and a point F on f. In this work the conditions for the canonical form of second order curve equation in an isotropic plane have been studied and all the exceptions have been discussed. Although conics in I2 have been investigated earlier, e. g. in the standard text-book of H. Sachs [8], or in the paper of Makarowa [7], this paper offers the method for second order curve classification in I2 based on Linear Algebra. The analogies with some known notions from Linear Algebra have been established and their mutual relations given. Furthermore, the invariants have been derived from the general equation of conic with regard to the group of motions in I2 which makes it possible for us to determine the type of a conic by means of arbitrarily chosen invariants without reducing it to canonical form. The obtained result is given briefly in the overview table. Such an approach can be understood as an example of classifying quadratic form in the spaces of various dimensions having no regular metric, e. g. quadrics in double isotropic space (I_3)^(2), pencils of quadrics in (I_3)^(2), etc.

conics; plane isotropic geometry

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