Development and Definition of Biquadratic Transformations Using Spheres and Two-Way Hyperboloids

Abstract

The insufficient use of quadratic transformations in applied geometry is explained by the fact that the methods of quadratic transformations are not developed much, although dozens of works by leading experts in applied geometry are devoted to the study of this problem, the development of graphic models, and their application in applied geometry. The research is devoted to the development of the theory of definition of biquadratic transformations of the plane. The essence of the proposed method for modeling biquadratic transformations of the plane, generated by a binary mapping of two surfaces of the second order, facilitation and solution of complex problems of applied geometries. And also, by means of graphical models of biquadratic transformations of the plane to facilitate the construction of curves of the second and fourfold orders. Considering a combination of non-linear surfaces of the second order, obtain subgroups of biquadratic transformations of the plane. The developed algorithm will make it possible to determine mathematical models of canonical biquadratic transformations of the plane, which is necessary for their practical application.