CORRECTING THE SHAPE OF A PLANE POINT POLYNOMA BY CHANGING THE POSITION OF ITS BASE POINTS

Abstract

Point polynomials are the foundation of compositional geometric modeling. The formation of point polynomials is carried out using a geometric interpolation method called compositional interpolation.

Compositional geometric modeling (CGM) is intended to create in a geometric way analytically (mathematically) formalized continual point sets, which are compositional models of geometric objects of arbitrary shape, reproducing the course of real processes and objects according to predetermined conditions, by performing compositional interpolation.

Since the point polynomial is an autumn-dependent parametric curve, its equation is formed with respect to the base points of the original discretely represented curve in the form of components, which are either non-harmonized or harmonized characteristic functions, which are a parametric basis of the corresponding point polynomials.

The article proposes to carry out the parametrization of the original geometric figure along the Ox axis, shows the definition of parameters at its base points.

In this case, without loss of generalization, test cases are calculated in the coordinate plane for six base points that define a point polynomial of the fifth degree. Shown in expanded form the construction of characteristic functions for a point polynomial, which is determined by six base points, is provided in general form, its equation.

It is shown that the possibility of constructing congruent curves, which is available in point polynomials, also contributes to a decrease in the number of computational operations in compositional geometric modeling. In this case, it is not necessary to calculate each new curve, but all calculations can be carried out on the originally calculated curve, and the result can be transferred to the corresponding congruent curves.

Keywords: compositional geometric modeling, geometric interpolation, discrete curve, point polynomial.