ON SOME DIFFERENCES IN THE FORMATION, NOTATION AND OPERATIONS BETWEEN ALGEBRAIC AND COMPOSITIONAL MATRICES

Abstract

The differences in the purposes of algebraic and compositional matrices are briefly formulated: these are, respectively, operations on linear systems and the analytical formalization of discretely represented irregular geometric objects. An explanation is provided as to why operations with algebraic matrices represent combinations, while the formalization of geometric objects obeys the properties of composition.

A definition of compositional matrices is given. Attention is drawn to the fact that the analytical representation of geometric objects by traditional methods is carried out relative to a pre-selected coordinate system, and conversely, in the compositional method of geometric modeling, discrete objects are analytically described relative to the points that define them. It is emphasized that it is precisely for this reason that the equations of geometric objects in the compositional method are proposed to be called — point equations. An example of the formation of a one-dimensional point compomatrix is provided, along with its general notation, its notation in element-wise and coordinate-wise representations. Whereas in analytical geometry, functions for the analytical description of geometric objects are formed relative to an optimally chosen coordinate system, in compositional geometry the analytical forms of geometric objects are formed using characteristic functions constructed relative to the object itself. It is noted that in our research two methods of parameterization are considered — along the accompanying polyline of the source curve and coordinate-wise. Based on the computed parameter values for each point, characteristic functions are formed, from which parametric compomatrices are composed, matching the previously created point compomatrices in size and arrangement of elements. Such a correspondence is natural, since both compomatrices are created for the same curve.

It is shown how the compomatrix of a curve as a geometric figure is formed through the product of the point and parametric compomatrices. The difference between the multiplication algorithms of compositional matrices and algebraic matrices is demonstrated. It is indicated that the multiplication of compomatrices is independent of the original coordinate system. Conversely, the multiplication of algebraic matrices arises as a result of geometric transformations with a change of the original coordinate system, that is, the multiplication of algebraic matrices represents the result of a coordinate system transformation. It is shown how the transition from the compositional matrix of a curve to the point polynomial that continuously describes the original discrete point sequence is carried out.

Keywords: compositional matrices, point compomatrices, parametric compomatrices, compomatrix of a curve, characteristic functions, multiplication of compomatrices, point polynomial of a curve.