THE CLASSIFICATION OF THE CONICS ACCORDING TO THEIR INVERSE IMAGES IN THE STEREOGRAPHIC PROJECTION

Abstract

An analytical geometry gives the affine and metrical classifications of the conics. Each class of the curves is characterized by the certain group of the invariants. This article deals with the technique which gives an opportunity to determine the class of the conic according to its inverse image in the stereographic projection. The concept of the stereographic projection is frequently used in the different branches of mathematics, and also in astronomy and geography. It is known that the images of the circumferences in the stereographic projection are always either circumferences or the straight lines. The aim of this article is the obtaining of the criteria, which an the opportunity to determine the type of the conic in the case when its inverse image in the stereographic projection is given. The formulae for the direct and inverse stereographic mapping have been obtained in the article. It has been shown that the inverse image of the conic on the sphere can be specified with the help of the system of the algebraic equations. One of the equations in this system is the equation of the sphere, and the left-hand side of the other equation is the homogeneous polynomial. The properties of the stereographic projection have been used for the formulating and the proof of the theorem on the particularities of the location of the points of the inverse images of the conics. The criterion, which gives an opportunity to determine the type of the non-degenerate conic in the case when its inverse image on the sphere is given, has been obtained. The similar criterion for the degenerate conics has been formulated. The fact that the coefficients in the equation of the conic and the coefficients in the equation of its inverse image are the same has been used essentially in the formulating and in the proof of these criteria. Hence, in order to determine the type of the image it is not necessary to know its equation. One can use only the equation of the inverse image. For these purpose, it is necessary to use the invariants of the conics. The examples, which show how the criteria work, have been given in the article.