Determination of Superposition Coefficients for Discrete Formation of Polynomial Functional Dependencies
Year: 2025
Type: article
Abstract: The shape control of a discretely represented curve (DRC) in the static-geometric method can be achieved not only by varying the functional external load but also through the coefficients in computational templates. These templates form the basis for constructing systems of finite-difference equations for DRC formation and indicate the proportional contribution of adjacent nodes to the desired formation. This article proposes a general approach to creating computational templates for modeling geometric objects (GOs) using superpositions of point sets. This aims to further study the influence of superposition coefficients, both arbitrary and of adjacent nodes of numerical sequences, on the formation of discrete analogs of elementary functional dependencies. One of the objectives of this study is to continue exploring the modeling of discrete geometric objects (DGOs) based on the classical finite difference method, the static-geometric method, and the geometric apparatus of superpositions. Since any polynomial of degree n is defined by n+1 points, determining the ordinate of any point given its abscissa requires substituting the coordinates of n+1 points into the polynomial function equation. This results in a system of algebraic equations containing n+1 equations and n+1 variables. Solving this system yields the polynomial coefficients a0 , a1 , a2 , a3 , … , an are found. In contrast to this approach, the recursive formula and the formula for determining the superposition coefficients proposed in this study allow for calculating the ordinate of any point of a polynomial of degree n given its abscissa without constructing and solving a system of n+1 equations. The ordinate of any curve point is determined as a superposition of the ordinates of n+1 points. In the proposed method of geometric curve modeling, the superposition coefficients are derived from systems of equations that contain one equation fewer than those used to calculate polynomial coefficients . Computational templates have been developed for the discrete formation of polynomial functional dependencies using superpositions of adjacent points’ coordinates. The approach presented in the article can be used to obtain expressions similar to formula (3) for calculating superposition coefficients for adjacent points of polynomials with two variables. Varying the superposition coefficients in the developed computational templates allows for studying the impact of these coefficients, both arbitrary and for adjacent nodes of numerical sequences, on the formation of discrete analogs of elementary functional dependencies. The shape control of a discretely represented curve (DRC) in the static-geometric method can be achieved not only by varying the functional external load but also through the coefficients in computational templates. These templates form the basis for constructing systems of finite-difference equations for DRC formation and indicate the proportional contribution of adjacent nodes to the desired formation. This article proposes a general approach to creating computational templates for modeling geometric objects (GOs) using superpositions of point sets. This aims to further study the influence of superposition coefficients, both arbitrary and of adjacent nodes of numerical sequences, on the formation of discrete analogs of elementary functional dependencies. One of the objectives of this study is to continue exploring the modeling of discrete geometric objects (DGOs) based on the classical finite difference method, the static-geometric method, and the geometric apparatus of superpositions. Since any polynomial of degree n is defined by n+1 points, determining the ordinate of any point given its abscissa requires substituting the coordinates of n+1 points into the polynomial function equation. This results in a system of algebraic equations containing n+1 equations and n+1 variables. Solving this system yields the polynomial coefficients a0 , a1 , a2 , a3 , … , an are found. In contrast to this approach, the recursive formula and the formula for determining the superposition coefficients proposed in this study allow for calculating the ordinate of any point of a polynomial of degree n given its abscissa without constructing and solving a system of n+1 equations. The ordinate of any curve point is determined as a superposition of the ordinates of n+1 points. In the proposed method of geometric curve modeling, the superposition coefficients are derived from systems of equations that contain one equation fewer than those used to calculate polynomial coefficients . Computational templates have been developed for the discrete formation of polynomial functional dependencies using superpositions of adjacent points’ coordinates. The approach presented in the article can be used to obtain expressions similar to formula (3) for calculating superposition coefficients for adjacent points of polynomials with two variables. Varying the superposition coefficients in the developed computational templates allows for studying the impact of these coefficients, both arbitrary and for adjacent nodes of numerical sequences, on the formation of discrete analogs of elementary functional dependencies.
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4408205098