ONE-DIMENSIONAL PARAMETRIC COMPOSITION MATRICES

Abstract

The article proposes a systematic description of one-dimensional parametric composition matrices, which serve as a formal tool for parameterising spatial discretely defined curves and subsequently constructing continuous composition lines. It is shown that each element of a one-dimensional point composition matrix corresponds to a unique characteristic function, which, together with other functions, forms the functional basis of a point polynomial that continuously interpolates the initial base points.

General forms of parametric composition matrices for three parametric directions are presented, along with a universal symbolic notation that ensures invariance with respect to the choice of coordinate system and enables extension of the approach to composition surfaces. An algorithm is proposed for constructing characteristic functions for each base point of a discretely defined curve, and its operation is demonstrated on an example of a curve with four points, including special cases of point coincidence.

An example is given of the formation of characteristic functions for the base points of a discretely defined curve, illustrating the rules of DDC (Discretely Defined Curve) point parameterisation and, directly, the generation of characteristic functions. It is stated that the set of characteristic functions for all base points of the DDC constitutes a functional basis of a point polynomial that continuously interpolates the initial base points. A general expression for this point polynomial is provided.

Variants of specific configurations of the initial base points are discussed, including those where points coincide. It is emphasised that the presence of such configurations does not require alternative algorithms for constructing characteristic functions. That is, characteristic functions are generated according to a unified method for both single and multiple points.

Furthermore, it is underlined that the formation of segments of composition surfaces requires the same number of points along each parametric edge, and this requirement is substantiated.

Keywords. Composite line, composite surface, composite parametric matrix, characteristic functions, point polynomial, functional basis from a point polynomial.