GLOBAL INTERPOLATION OF THE POINTING POLYNOMIAL OF THE GEOMETRIC COMPOSITION WITH MULTIPLE POINTS

Abstract

The article shows the sequence of parameterization, along the coordinate axis, of the original discretely presented line (DPL) and is presented in general form by a point polynomial. Possible options for the appearance of multiple points are considered and the values of the parameters for these options are presented. It is indicated that with the appearance of multiple points on the DPL in the constituent elements of a point polynomial, uncertainties arise. It is proved that all these uncertainties are revealed, the limits of which, at the nodal points, are zero or one. It is shown that the uncertainties that arise with the appearance of multiple points on the DPC are not an obstacle to global interpolation using a point polynomial. That is, for any composition of three points, the construction and recording structure of a point polynomial remains unchanged. However, there are no restrictions on creating a composition of three points. This fact is proved in this article. A composite numerical matrix is presented in accordance with which conditional interpolation occurs. The elements of this compositional matrix are the values of the characteristic functions of the interpolant at the nodal points. It is shown that the elements of the compositional interpolation matrix do not change in the presence of any geometric composition of three points. Only the status of these elements can be changed. In one case, their values are accurate, and in the other, they can be the limit to which the value of the characteristic function of a point polynomial follows.

Geometric modeling of volumetric objects of arbitrary shape requires the construction of its surface. Usually, the construction of surfaces is done by drawing a mesh on it. If a mesh is applied on the surface of a geometric body of an arbitrary shape, has a constant number of lines in the direct and transverse directions, then cells of various sizes, both large and very small, will appear. On large cells, the surface reproduction error will increase, and on small cells, the consumption of modeling resources will increase, and the efficiency and quality of modeling will decrease.

Keywords: multiple points, geometric composition, compositional matrix, disclosure of uncertainties, point polynomial.