EXAMPLES OF FORMATION OF CHARACTERISTIC FUNCTIONS FOR A ONE-PARAMETER POINT POLYNOMIAL

Abstract

Based on previous research, it is established that one of the key conditions for constructing point polynomials is the formation of one-parameter characteristic functions, from which a one-dimensional compomatrix is created, serving as the functional basis of the point polynomial. It is emphasized that for each point of the source discretely defined curve, a separate characteristic function is formed. For this purpose, each point of the source point sequence is parameterised along the links of its accompanying polyline.

An explanation is provided regarding the formation of characteristic functions in the form of simple ratios of three points, which is an invariant of parallel projection. An example of parameterisation of base points of a discretely defined spatial curve is given.

Examples are provided of forming characteristic functions for a straight line, for three characteristic functions passing through a curve defined by three points, and for four characteristic functions for a curve defined by four base points. Graph-schemes are created for three and four characteristic functions. Generalised expressions are given for all considered characteristic functions and for a discretely defined curve with N base points. It is stated that all characteristic functions together constitute, in parametric form, the functional basis of a point polynomial that continuously interpolates the source point sequence. Equations of the point polynomial in general and coordinate forms are provided.

As a result of the study, it was established that the construction of a continuous composite spatial curve defined by a discrete set of points requires the formation of characteristic functions of the basis points, which constitute the functional basis of the curve and are elements of a one-dimensional parametric composite matrix. The proposed generalized representation of the characteristic function makes it possible to avoid the indeterminate form zero divided by zero and ensures its correct construction.

Keywords: рoint polynomials, one-parameter characteristic functions, parametric compomatrix, functional basis of point polynomial, parameterisation of points, discrete curve.