ONE-DIMENSIONAL COMPOSITION DOT MATRICES

Abstract

The article substantiates the theoretical foundations for constructing point composition matrices as a means of continuous formalization of spatial geometric objects (point, line, surface, solid) within the framework of point calculus. It is noted that point composition matrices provide an effective approach for describing discretely defined geometric entities, especially one-dimensional lines, and enable the development of generalized mathematical models based on their parametric representations. Examples of notations and general forms of point composition matrices are presented for spatial lines and lines lying in planes, taking into account three main parametric directions (U, V, W). Special attention is paid to the analysis of the general form of such matrices and the transition to coordinate-based notation in the Cartesian coordinate system. The structure of composition matrices for each parametric direction is defined, and the expansion of such a matrix is illustrated for an n-dimensional parameter space, where the parameters reflect the evolution of processes associated with a given geometric line. An alternative approach to interpreting these matrices is also provided: for each discrete point of the space, the values of all parameters characterizing the state or process at that point are revealed separately, allowing for a deep local analysis. Unlike traditional algebraic matrices, composition matrices are not limited to linear structures and can contain elements of various natures—numerical, symbolic, functional, or even other matrices. Considerations are offered regarding the criteria for recognizing a composition matrix as an independent geometric object. Coordinate composition matrices in three-dimensional and n-dimensional parameter spaces are also described, serving as tools for conducting calculations within the domain of compositional geometry. The proposed methodology may be effectively applied to the modeling of complex multifactor geometric systems, where both precise structural representation and analysis of dynamic or functional characteristics of the object are required simultaneously.

Keywords. Point composite matrices, geometric objects, coordinate composite matrices, one-dimensional composite matrices.