POINT POLYNOMS AS COMPOSITE GEOMETRIC MODELS

Abstract

The definition of point polynomials is given, it is indicated that their equations are irrelevant to the original coordinate system, but are formed relative to the base points of the discrete curve on the basis of which this point polynomial is formed. In its general form, the equation of a one-parameter point polynomial is written. The definition of one-parameter characteristic functions and their expressions in a general form is provided, the sequence of parametrization of curved lines for determining parameters at base points with the aim of their application for calculating characteristic functions is shown. The sequence of transformation of characteristic functions into BN-coordinates of one-parameter point polynomials is given, which is called harmonization of characteristic functions. With the use of BN-coordinates, in a general form, the equation of the harmonized point polynomial is given. The advantages of using harmonized point polynomials over non-harmonized ones are indicated. The equation of a two-parameter point polynomial and its characteristic functions for all base points along both parametric directions U and V are provided, in general form. It is emphasized that each base point of the original discretely presented surface is chosen only at the intersections of the line frames formed along the two parametric directions. Because of this, each base point has two characteristic functions, that is, it is determined by two coordinates in accordance with each of the parametric directions. A method of calculating the values of parameters at all its base points, which are used, in the future, for compiling expressions of characteristic functions, has been developed, in a general form, for two parametric directions. It is emphasized that the implementation of the operations of multiplication of characteristic functions and BN-coordinates between themselves and on base points for the compilation of point polynomials is best carried out in compomatrix form. In a general form, examples of multiplication of parametric compomatrixes and multiplication of a point compomatrix by a parametric one are provided. At the same time, it is emphasized that the operations of multiplication of two compomatrices are carried out only between their elements, which have the same either single or double indices. Attention is drawn to the peculiarities of the harmonization of two-parameter characteristic functions and point polynomials. A general view of the harmonized two-parameter point polynomial is provided.