COMPARISON OF COMPOSITE AND TRADITIONAL FIRST DERIVATIVE POINT POLYNOMIALS IN THE PROCESS OF THICKENING THEIR OUTPUT FLAT DISCRETE CURVES

Abstract

A segment of a discrete flat curve line consisting of four basic points is considered. By using the Cartesian coordinates of these base points, a corresponding point polynomial is formed in a parametric form, for which, according to the Newton-Leibnitz differentiation methods, the traditional first derivative is formed and its values are calculated at several current points.

An explanation of the difference between the traditional and composite first derivatives is given, which is that the traditional derivative is formed by differentiating the characteristic functions, while in the composite derivative the characteristic functions of the point polynomial remain unchanged, its base points are replaced by the values of the derivatives at these base points. For the same parameter values, the values of the traditional and composite derivatives are calculated, then their comparison is carried out, which showed that at all current points these values of the first derivatives differ by the amount of the calculation error, and at the base points they are equal to each other.

Next, the first thickening step is carried out, according to this step, the parameterization of the base points and thickening points is performed. According to the results of the parameterization, a point polynomial is formed for the first step of thickening. By differentiating which we find the equation of its traditional first derivative and calculate its values at the base and current points. The found values of the traditional first derivative are substituted into the point equation of the composite first derivative and its values are calculated at the same current points. The values of traditional and composite derivatives are compared, which also differ from each other by the amount of computational error, that is, they are equal to each other.

Similar actions are carried out for the second step of thickening and the same result is obtained, that the values of the composite and traditional derivatives at the same current points are equal to each other, if the values of the traditional first derivative are substituted into the point equation of the composite first derivative, instead of base points.

As a result of the conducted research, it was concluded that the composite derivative is more generalized in relation to the traditional Newton-Leibnitz derivative. In addition, by using the diff projection band, by thickening the flat original discrete curve, the composite derivative approaches and coincides with the traditional derivative when the diff projections of the composite derivative at the base points coincide with the values of the traditional first derivative.

Key words: point polynomial, composite derivative, traditional derivative, diffraction band, discrete curve thickening.