Three-Dimensional Composition Matrices and Their Applications for Creation of Compositional Geometric Models of Volume Objects of Any Arbitrary Form

Abstract

geometric figure of arbitrary form is given.Keywords: three-dimensional compomatrices, composite model, harmonized point polynomial, geometric method of interpolation, basis state, types of compomatrices, designation of compomatrices, compomatrices point, compomatrices parametric, compomatrices coordinate. Statement of problem.Unification of each geometric figure (GF) is by dividing it into two components: geometric and parametric.The geometric component of the unified GF is presented in the form of point matrices in parametric directions.The parametric component of the unified GF is presented in the form of parametric compomatrices. An algorithm for the formation of parametric threedimensional computer matrices is provided.It is emphasized that all calculation operations are carried out through the use of three-dimensional coordinate matrices (calculated), which are compiled according to the scheme of the corresponding point matrices.Emphasis is placed on the fact that in this study a compositional model of a segment of a geometric body consisting of three points in each of the parametric directions is built. It is also pointed out that the proposed algorithm will be true for creating models of segments of three-dimensional geometric shapes that hold more points in each of the three directions that indicate the size of this segment.Research method. In this work, the research is carried out by the methods of compositional geometry (CG) [13] with powerful tools for the formation of point polynomials that perform the initial conditions and compositional matrices.CG is based on the point calculus of Balyuba-Naidysh (point BNcalculus). [1] Geometric methods of interpolation with one-, two-and three-parameter point polynomials are used in CG methods.Geometric method of interpolation (GMI). Option 1. It is provided by point polynomials, which have their components of the product of each of the basis points for the corresponding, specially formed, characteristic functions, and is that each of these characteristic functions is equal to one only at the basis point, which is its multiplier and through which , for the corresponding value of the parameter t, is a point polynomial, and at other basis points, this characteristic function is zero. That is, with the geometric method of interpolation in each interpolation node, a point polynomial has only one term equal to the corresponding basis point, thus ensuring global interpolation of all source basis points.Option 2. Provided with characteristic functions that are formed based on the geometric conditions laid down in the original geometric composition, and are components of point polynomials, which in the interpolation nodes are zero or one, due to which global interpolation of starting points geometrically. GSI