Foci of Algebraic Curves

Abstract

Curves have always been part geometry. Initially, there
 were lines and circle, then it was added to a conic section and
 later, with the advent of analytic geometry, they added more complex
 curves. Particularly in a number of lines are algebraic curves
 that are described by algebraic equations. Curves found application
 mostly in mechanics. Today algebraic curves used in engineering
 and in mathematics, in number theory, knot theory, computer science,
 criminology, etc. With the bringing to account of complex
 numbers became possible to consider curves in the complex plane.
 It has expanded the horizons of geometry and enriched their knowledge
 on curves, particularly on algebraic curves. Our goal is to give
 a geometric picture of the foci of algebraic curves clearly show the
 position of the foci in the plane, show how the number of foci associated
 with a class curve. The solution of this problem we see in
 the application we have developed ways to visualize imaginary
 images to the study of foci and focal centers of algebraic curves.
 This article explains the concept of the foci of algebraic curves
 shows the basic principle of the curve-theory and offers a method
 for the identification of the foci. The geometric picture of the foci
 is shown in a diagram, which is putted together from two tables.
 One table shows the real curve with her foci, the other table shows
 an imaginary cut of the curve, on which the isotropic line contacts
 the cut and under them intersects in a real point. The point is a
 focal point of the real curve. This project shows 16 diagrams for
 conic, cubes and quadrics.

Design of the Third-Order Curves of the Type the Hyperbolism of the Hyperbola

“…Let us have a parabola 2 y ax bx c    , with a vertex at the origin, c = 0, and its focal axis coincides with the ordinate axis (figure2) [4,6,7]. The straight lines of the beam P and the straight lines of the beam L form a bundle of first-order straight lines (figure 3).…”