A single constructive algorithm for constructing foci of second-order curves

The article is devoted to the analysis of some geometric schemes and discussion of the issues arising in this connection of the theory of constructing second-order curves by methods of constructive synthesis. The article shows that the currently used definitions of the center of the second-order curve and the diameters of these curves conflict with the principle of indistinguishability of conics in projective geometry. The ways of eliminating these contradictions are proposed and a unified algorithm for constructing foci of second-order curves is developed on their basis. The author's reasoning, based on the apparatus of projective geometry, will reveal a number of contradictions in the currently existing definitions relating to second-order curves, and their elimination will provide an opportunity to develop a unified approach to the construction of some geometric images initiated by second-order curves and give them a general constructive justification. As a result of the analysis of geometric schemes, a number of concepts of projective geometry were clarified, which made it possible to unify the solution of problems related to the construction of focal points of second-order curves. A unified algorithm for constructing all four foci of the second-order curve is presented. Thus, the basis has been laid for expanding the fields of application of geometric models to imaginary geometric images covered by the concept of a "second-order curve", and conducting research on the resulting geometric images and schemes.

geometric modeling, second-order curve, conic, focus, Simplex, imaginary image, collineation, improper point, asymptote, cyclic points