A Unified Constructive Algorithm For Second- Order Curves’ Foci Creation

Abstract

While using conventional tools for solving geometric problems, it is difficult to obtain and analyze results where imaginary geometric images appear. Despite the recognition of legitimacy and scientific value of imaginary solutions presenting in geometric constructions, the question on such solutions’ appropriateness and practical feasibility remains no completely clear up till now. That’s why, for most practitioners imaginary solutions are presented as something unattainable or unimportant. However, the introduction of imaginary geometric images into the practice of geometric modeling makes it possible to obtain solutions in an exhaustiveness, to develop unified algorithms for solving problems that were usually presented as either not solvable or reduced to solutions in partial settings. The use of computer technologies and the paradigm of constructive geometric modeling allow eliminate this problem’s acuteness, and direct efforts both at geometric theory’s improvement and introduction of scientific achievements in this area at the field of practical applications. Automation means for geometric experiment make it possible to find new regularities in seemingly well-known mathematical facts, to come to more general understanding of geometric concepts and images. This paper is devoted to analysis of some geometric schemes and to discussion of arising from it questions related to the theory of second-order curves creation by the methods of constructive synthesis. In the paper it has been demonstrated that the currently used definitions of second-order curves’ center and diameters contradict the principle of conics indistinguishability in projective geometry. The ways for eliminating of these contradictions have been proposed, and a unified algorithm for the second-order curves’ foci creation has been developed based on these ways.