ONE AND TWO-DIMENSIONAL VALID COMPOSITION MATRICES

Abstract

The definition of compositional matrices (compomatrixes) is given, mathematical objects are defined that can be elements of compomatrixes. Shown are the requirements for indexing one-, two-, and three-dimensional compomatrixes and their purpose. The symbol designation of the real compromomats is established and it is indicated that they are intended for the analytical formalization of the description of geometric figures. It is indicated that the need to introduce the concept of “composite matrices” is caused by the nature of the formation of geometric figures (GF). It is determined what is the unification of the GF and why it is needed in compositional geometric modeling. The rules for the formation of a point compomatrix and a parametric compomatrix are provided, and their conventions are defined. It is proved that compomatrixes are used for geometric modeling of objects, each point of which is K-valued, that is, defined by the k-coordinates of the parameter space. It has been established that the number of elements and the form of their recording in the compatrix are in complete accordance with the number of points and their location on the original GF.

A representation of the geometric model of the initial HF is provided, in a complimentary form, and examples of its creation are given for one- and two-dimensional compatatrices.

The zero and single compomatrixes and their notation are defined. Also shown are the formations and designations of the numeric compomatrix. Recording rules and notation for calculated (coordinate) compomatrixes are provided.

The sequence of transition from the compatrix form of a GF to a point polynomial, which is an interpolant of this GF, is shown. A compatrix is ​​proposed for the interpolant, which is written in expanded form, and the sequence of its separation into geometric and parametric components in the form of the corresponding compomatrix is ​​shown. It is proved that the pointwise interpolant compomatrix is ​​composite, and the parametric is combinative. The analysis of the parametric compatrix for point polynomials is carried out, its trace and determinant are given importance, the features of the transposed compomatrix to the original parametric are indicated. The main feature of the transposed parametric compomatrix is ​​that it is equal to the original compomatrix, which allows the mobile simplex method to be used without restrictions for the geometric modeling method.

Keywords: one-dimensional and two-dimensional composite matrices, compomatic equations, point polynomial.