In the article we examined the origin of the name of the dome and its connection with the sunny surface of a yurt. The new design of the dome surface is also shown. In this study, using a graphical model of the biquadrate transformation, each point of the transformed line, i.e. n of the prototype, is transformed into points forming a cross-section of the dome surface, while the transformed line gives a cross-section of the surface of the Sun. Then we rotate this section of the found dome surface relative to the vertical section Oх2and get the desired dome surface. When designing a removable dome surface, the format and dimensions of the section are set in accordance with the preliminary requirements. Thus, the biquadrate transformation method that we propose when designing a section of a domed surface, that is, a project of a domed curved surface, is practical and simple for designers. The proposed method in its simplicity is designed to solve various similar curved surfaces and engineering problems.

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.