ANALYSIS OF GEOMETRIC FIGURES USING ONE AND TWO-DIMENSIONAL COMPOSITE MATRICES

Abstract

It is determined that operations on compositional matrices (compatrixes) are carried out through operations on their elements and in a certain correspondence with geometric transformations that are applied to geometric figures (GF), which are described by these compatrixes.

It is proved that for one-parameter HF (geometric figures) one-dimensional compomatrixes can be ordered both in rows and columns, designations of one-dimensional point and parametric compomatrixes are provided.

It is analyzed that for one initial GF with different algorithms for its parameterization, different solutions will be obtained - parametric compomatrixes.

The conditions for the identity of two one-dimensional compomatrixes constructed for one initial GF and for congruent GF are determined.

We consider the rules for compiling and designating two-dimensional real compomatrixes for the initial quadrangular and triangular geometric figures. The conditions for the appearance of empty elements in them are indicated, the rules of operation with empty elements are justified.

The examples show that the outline of the recording of the actual elements of the compomatrix coincides with the outline of the segment of the original GF for which this compomatrix is composed, while the indicated applies to both point and parametric compomatrixes.

Using the method of moving simplex, an example is given of the formation of a world-wide compromise of a geometric figure, the choice of its size and the preparation of the point equation of a point polynomial based on it are interpolated, which interpolates the original GF for which this world-wide compromise is compiled.

It is also determined that even when the outline of the entries of the real elements of the dwarven compomatrix is ​​not rectangular, it is still considered rectangular in size, which is determined by the largest number of elements in the column and row.

The signs of equality of two two-dimensional compomatrixes are established. It is established that they will be equal only if they are composed for one initial geometric figure with one algorithm for parameterizing its components. The signs of congruence of two geometric figures by their two-dimensional compositional matrices are also established.

Keywords: point polynomial, compositional matrices, unification of geometric figures, characteristic functions, geometric interpolation method.