# Harmonization of point polynomials.

• V. Vereshchaha Melitopol State Pedagogical University named after Bohdan Khmelnytsky
• –Х. Adoniev Zaporizhia National University
• –Ю. Pavlenko Melitopol State Pedagogical University named after Bohdan Khmelnytsky
• –Ъ. Lysenko Melitopol School of Applied Geometry named after Volodymyr Naidysh
Keywords: Point polynomials, harmonization, characteristic functions, BN coordinates, B-curves.

### Abstract

¬†A point polynomial is an entire rational function in parametric form, which consists of the sum of products in which the first factors of each of the terms is the base point of the original discretely presented line (LPL), and the second is the algebraic factor, which is a whole rational expression that is served in the form of the product of the difference between the parameters of the corresponding nodal points and the current parameter - the argument t for the intermediate point. Point polynomials form the basis of compositional geometry and the compositional method of geometric modeling. Compositional geometry is a geometry in which each initial geometric figure (GF) is divided into geometric and parametric components and any problem is solved with respect to all the base points of this GF, regardless of the coordinate system in which these base points are defined. The process of dividing the GF into geometric and parametric components was called by us - the unification of the original GF. The geometric component is described using the point matrix - AT, and the parametric - using the parameter matrix - AP. The components of a point polynomial - terms, are the products of the corresponding elements of the compositional matrices - point AT = ((Aij)) and parametric AP = ((aij)). Point composite matrices describe the geometric composition of points for a certain number of points. Moreover, there are absolutely no restrictions on the coordinates that these points determine. That is, changing or replacing any of the points of the geometric composition, or even the entire composition of the points, as a whole, will only lead to a change in the elements of the composite matrix (CM) of the point, and will not entail a change in the further solution. At the same time, there will be absolutely no changes in the parametric CM, which determines the mutual arrangement between the elements of the geometric composition of the points that form the GF. Except when the newly introduced points changed their location along the direction in which the elements of the initial GF were parameterized. And even in this case, only certain elements of the parametric CM are subject to change, and the further decision algorithm and the solution itself are not subject to change at all. Under the composition, in general, it is necessary to understand a discrete set of interconnected elements (parts, objects, factors, points, etc.) that make up the whole object, which is perceived as a whole, has a certain internal unity, while changing or replacing any of of these elements, in general, does not entail any changes for the remaining other elements of the existing geometric composition. A geometric composition is a composition whose elements are a nonempty finite set of points, some of which can form a certain subset, and at the same time, for each element of this set its own sizes and dimensions that determine their relative position are established.

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