GEOMETRIC ASPECTS OF TRANSCENDENT TRANSFORMATION OF SPACE

Abstract

This article discusses issues related to the study of the capabilities of some special coordinate systems that can be used in the design of surfaces of complex curvilinear shape. Curved surfaces are used in many branches of science and technology, including mechanical engineering, construction, architecture and other fields of knowledge and production. The design of complex curved surfaces can be significantly simplified if the geometric apparatus for creating a special coordinate system is used in the design. In such cases, the geometric apparatus of a special coordinate system is organically linked with the geometry and kinematics of the constructed surface. In the practice of architectural design, there are many examples of the use of a special coordinate system in the design of shells and various curvilinear options for coatings of building objects and other structures. In this regard, this paper offers a detailed description of the geometric transformations of a rectangular Cartesian coordinate system into other coordinate systems. Any three-dimensional coordinate system is represented in the form of three conditional axes and three quantities plotted on these axes. The axes can be rectilinear or curvilinear, and the coordinates can be linear quantities, angular, expressed as a prime number or in general by some function of some predefined parameters or any function at all. Any point, line, or even surface can be used as the origin of the selected coordinates. Thus, the resulting set of coordinate systems can be called a generalized coordinate system. At the same time, the essence of any spatial coordinate system can be represented by a certain congruence. The geometric apparatus of a generalized coordinate system is any congruence of straight or curved lines, taking into account the specific conditions connecting the parameters of the congruence. The definition of "generalized coordinate system" includes the well-known cylindrical and spherical coordinate systems.