Kassel', Germany
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.
algebraic curve, circular curve, order of the curve, class of the curve, conics, cubics, quartics, isotropic lines, cyclical points, ideal line, imaginary section, tangent line, ordinary foci, special foci.
Введение
Кривые линии всегда были частью геометрии [1]. О кривых линиях статьи имеются в различных научных сборниках [6; 13–15; 18; 20], в том числе и в журнале «Геометрия и графика» [6; 15]. Вначале это были прямые и окружности, затем к ним добавились конические сечения и позже, с появлением аналитической геометрии, – более сложные кривые. Особо в ряду линий стоят алгебраические кривые, описываемые алгебраическими уравнениями. Кривые линии находили приложение большей частью в механике. Сегодня алгебраические кривые используются как в технике, так и в самой математике – в теории чисел, теории узлов, информатике, криминалистике и др.
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