THE FINITE DIFFERENCE DEPENDENCIES AND THE SUPERPOSITION COEFFICIENTS OF TWO POINTS OF DISCRETE CURVES

Abstract

The discrete models’ formation of n-dimensional geometric images (GO) involves using of methods that require the application of significant computing resources. A geometric object of an arbitrary shape can always be represented by an ordered set of points according to a certain law so that the coordinates of any point inside the contour (area) can be determined. The only question is the necessary density of the source information and the costs of its acquisition, processing, and storage.

Consequently, it is necessary to conduct research on new methods of formation (GO) that allow ensuring minimum costs for obtaining results. The use of the geometric apparatus of superposition in the formation of discrete models (GO) by the static-geometric method allows determining the coordinates of nodes of discrete frames based on the coordinates of the minimum number of specified nodes without compiling additional large systems of linear equations.

The aim of this work, in particular, is to study the method of forming discrete images of curved lines based on the classical method of finite differences, the static-geometric modeling method, and the geometric apparatus of superposition.

The article proposes a technique for deriving analytical dependencies for discrete values determining the superposition coefficients of two given nodal points and the shape-forming value of the finite difference for modeling one-dimensional geometric images based on a given symmetrical calculation scheme.

This procedure can be applied to derive similar analytical dependencies, which allow determining the values of the superposition coefficients of two given nodal points on the basis of any numerical sequences and arbitrary calculation schemes and, thus, solving problems of continuous discrete interpolation with one-dimensional numerical sequences of a wide range of elementary functions.

Keywords: discrete modeling, static-geometric method, geometric apparatus of superposition, finite difference value, superposition coefficients.