About One Way of Forming Conics

Abstract

In the articles of the Geometry and Graphics magazine devoted to the properties of the Dupin cyclide, the construction of a conic – ellipse, hyperbola and parabola – using the properties of the cyclide was considered. At the same time, the center of the transformation was located on a straight line connecting the centers of the two base circles, and its location on such a straight line was negotiated separately and was located as the center of homology. To construct a parabola, it was necessary to take a straight line instead of the second circle, and the center of the transformation – the center of homology – had to be located at the intersection point of a straight line passing through the center of the first circle perpendicular to the second circle-a straight line with the first circle. Two different parabolas were obtained as a result of the transformation. In this paper, it is proved that if we take the center of correspondence that does not belong to a circle, we get other second–order curves - ellipses and hyperbolas. The construction of an ellipse is geometrically proved. To do this, the center of correspondence must lie on a straight line connecting the centers of the circles, but outside the actual circle. Several examples are considered. If the center of correspondence is inside the circle, we will have a hyperbola. Thus, having initially given only one con-figuration from a straight line and a circle, it is possible to obtain all conics: ellipses, parabolas, and hyperbolas, passing into one another. The proposed scheme for constructing conics can be used for computer drawing of all conics, which is more convenient than with the available options sewn into today's graphical drawing systems.