GEOMETRIC METHOD OF INTERPOLATION OF POINT POLYNOMIALS IN PARAMETRIC FORM

Abstract

The geometric interpolation method provides global interpolation of discretely presented lines (DPL) and, at the same time, does not use systems of algebraic equations to find the interpolant coefficients, and there is no need to find interpolation coefficients, since the point log does not provide for this.

A point polynomial is an entire rational function, consists of a sum of products, the first factor of each of the terms of which is the point of the original DPL, and the second is algebraic factors in parametric form, they are entire rational expressions that are represented as the product of the difference between the parameters of the corresponding nodal points and the current argument argument t.

The parametrization of the initial DPL can be applied along the coordinate axis, either along an arbitrary straight line, or along the length of the cells connecting in a row all points of the original DPL.

The parametric compositional matrix, whose elements are the algebraic factors of the components of the point polynomial, is the parametric component of the unified geometric figure of the initial DPL, provides global geometric interpolation, bypassing the coefficients, compiling and solving a system of linear equations.

The unification of the initial geometric figure provides for its division into geometric and parametric components. The geometric component is described using the point matrix, and the parametric component is described using the parameter matrix.

The components of a point polynomial are terms that are the products of the corresponding elements of the composite matrixes of point and parametric. The geometric component is represented directly by the points of the initial DPL, and the parametric - by whole rational expressions in the form of the product of various parameters at the nodal points of the original DPL and the current parameter.

The point interpolation polynomial in the method of constructing elements and the geometrical sense of functioning is like an interpolation polynomial in the Lagrange form. However, it is much more powerful due to the fact that with the help of a point polynomial the solutions occur in the coordinate space as a whole, and not separately on each of the coordinate planes. In addition, the resulting solution in space can be transferred to any of the coordinate planes or even to subspaces.

Another advantage of a point polynomial is that its record does not need to be changed if there are multiple points (coinciding) on the original DPL. Two-, three-, n-fold points arise on grids of volumetric objects of arbitrary shape.

Keywords: point polynomial, compositional matrix, unification of a geometric figure, multiple points.