ДОСЛІДЖЕННЯ ЗАКОНОМІРНОСТЕЙ ЗМІНИ ВЕЛИЧИН КОЕФІЦІЄНТІВ СУПЕРПОЗИЦІЇ ОДНОВИМІРНИХ ФУНКЦІОНАЛЬНИХ ЗАЛЕЖНОСТЕЙ НА ПРИКЛАДІ ПОЛІНОМІАЛЬНИХ ФУНКЦІЙ

Abstract

In modern conditions, in design of structures, networks, products, the stage of constructing and analyzing geometric models of objects, processes and certain phenomena takes a significant place. An important problem is a creation of new methods for constructing lines and surfaces of geometric images, which fully corresponds to the purpose of automated design and reproduction. The static-geometric method of discrete geometric modeling of curved lines and surfaces makes it possible to obtain discrete frames of curved surfaces under the action of an external shaping load and, moreover, is simple and intuitive. Usage of the static-geometric method allows also obtaining discrete frames of curved surfaces on an arbitrary bearing contour. At the same time, it should be noted that the apparatus of this method is based on solutions of cumbersome systems of linear equations, which does not provide information on geometric properties and features of local parts of models.  In the article it is proposed the use of the geometric apparatus of superpositions in combination with the static-geometric method of discrete geometric modeling, which can significantly increase efficiency and expand capabilities of the discrete modeling process of geometric images. On the basis of the geometric apparatus of superpositions, regularities of values change in superposition coefficients of three arbitrarily given nodal points of polynomial functions are investigated. These researches determine a general approach to obtaining similar regularities of values change of superposition coefficients of three arbitrary given nodal points (as adjacent as not-adjacent) to determine coordinates of n points of any modeled one-dimensional functional dependencies and arbitrary one-dimensional sets of points. Using polynomial functions as an example, it is shown that the obtained formulas for calculating values of superposition coefficients of given three nodal points for selected design schemes allow solving problems of continuous discrete interpolation and extrapolation by numerical sequences of any one-dimensional functional dependencies (as determining ordinates of desired points of discrete curves by three given ordinates of nodal points) without time-consuming operations of composing and solving large systems of linear equations. Keywords: discrete modeling, geometric images, static-geometric method, geometric apparatus of superposition, polynomial functions.