SYSTEMATIZATION OF INTERPOLATION METHODS IN GEOMETRIC MODELING

This article proposes to show the connections between the practical problems of geometric modeling, for the solution of which there are no adequate interpolation methods. Interpolation methods in mathematics first emerged as part of mathematical modeling, and eventually became a separate area of mathematics. The classical definition of interpolation is the constructive reproduction of a function of a certain class according to its known values or the values of its derivatives at given points.Interpolation problems arise in spaces of different dimensions when solving engineering problems. Elements of such spaces can be not only points but also other geometric shapes. The dimension of space is always equal to the parametric number of the element of space. In applied geometry, interpolation problems have been studied since the 1960s. Since there are so many publications related to interpolation problems, systematization is proposed for their analysis. The largest number of existing publications is devoted to the problems of one-dimensional interpolation in two-dimensional space, both continuous and discrete. A much smaller number of publications are devoted to two-dimensional continuous and discrete interpolation. The variety of interpolation methods is due, first, to different sets of initial geometric conditions. Only in some publications the choice of initial conditions depending on the purpose of interpolation problems in specific sectors of the economy. In many publications, the choice of interpolation method is related to the features of the basic mathematical apparatus. Among the considered publications there is none where the influence of parameters of the set points, namely distances from them to the current point of an interpolant, on its parameter is directly considered. Such interpolation can be the basis for modeling various energy fields, where given points are point sources of energy.