THEORETICAL FOUNDATIONS OF THE POINT CALCULUS AS A MATHEMATICAL APPARATUS FOR GEOMETRIC AND COMPUTER MODELING

Abstract

The purpose of the article is to acquaint the reader with the new mathematical apparatus – point calculus, which was developed for solving engineering and scientific problems by the leading scientists of the Melitopol School of Applied Geometry and continues its active development in the works of Russian and Ukrainian scientists-geometers. The article presents the fundamental definitions and terms, basic methods, metrics, fundamental theorems of point calculus, shows the principles of parametrization of a straight line, plane and three-dimensional space in point calculus. Possibilities of point calculus include the definition of continuous and discrete geometric models of objects, processes and phenomena in the form of point equations and computational algorithms based on them. The advantage of the new mathematical apparatus is the representation of geometric objects in the form of a set of projections on the axes of the global coordinate system, which makes it possible to determine the geometric models of objects, processes and phenomena in spaces of any dimension. The use of point calculus in computer graphics and virtual reality, solid modeling of isotropic and anisotropic bodies and computer modeling of objects consisting of nanoparticles is seen as promising. Therefore, the main direction of promising research is seen as the further expansion of theoretical and applied point calculus tools, and its popularization by solving a wide range of important engineering and scientific problems of an applied nature.