Analytical and computer modeling of surfaces with the curvilinear projection method

The article discusses the shaping of surfaces by analytical methods and their visualization by means of computer graphics. . This topic attracts the attention of architects, engineers and mechanical scientists, for whom it is important to see the surfaces in the structures of machines for various purposes and in the forms of structures, to approximate complex surfaces with simpler ones - analytical ones; choose a rational form of the shell from several options, taking into account the functional, technological and ergonomic requirements for the product. The purpose of the work is to study the methods of formation of surface shapes. The research method is the general analytical theory of applied surface shaping, which meets modern requirements for the use of computer technology. In this paper, parametric equations of the cyclic surface of Joachimsthal are obtained, the possibilities of shaping such surfaces, which are recommended for practical use, are shown. Using examples of visualization of surfaces by means of computer graphics (Maple program), the use of analytical models is shown, which makes it possible to quickly and reliably assess the constructive and aesthetic qualities of the shell in scientific research, design, and manufacture. The ways of developing an analytical apparatus are determined, which allows modeling the process of curvilinear projection and the formation of surfaces as a system of projecting rays passing through a given surface projection. The purposeful choice of the parametric form of analytical modeling of surfaces contributes to the direct use of models in automated design systems, pre-production systems and in modern computer graphics packages (Kompas 3D, Renga, Revit, Ansys, Lira Sapr, Scad, etc.).