DISCRETE INTERPOLATION OF TWO-DIMENSIONAL POINT SETS BY FOUR POINTS SUPERPOSITIONS FOR PARABOLIC SURFACES

Abstract

In the article it was proposed the general approach to determine change patterns of superposition coefficients of two-dimensional point sets. This makes possible to solve problems of continuous discrete interpolation and extrapolation any two-dimensional functional dependencies by numerical sequences, using four arbitrary nodal points.
 Coordinates of any point of two-dimensional point set can be represented by a coordinates’ superposition of four arbitrary points from this set. This allows to get analytical dependences to determine superposition coefficients values, solving corresponding systems of equations.
 It was investigated the discrete analogues formation process of two-dimensional geometric images, using as an example surfaces, whose components are polynomial functional dependences.
 In the research we established change patterns of superposition coefficients values of three nodal basic contour points and an internal nodal point. These patterns are represented in the form of surfaces-graphs of two variables numerical sequences for a chosen calculation scheme.
 The received regularities allow to form surfaces, using the applicates of three basic contour points and an internal point. These surfaces are on a given calculation scheme and components of their frames are polynomial functional dependences.
 The studies determine the general approach of obtaining similar change patterns of four arbitrary points superposition coefficients. These points can be as adjacent as non-adjacent node points of a selected calculation scheme. The regularities are used to determine n points coordinates of any two-dimensional functional dependences and arbitrary two-dimensional point sets.
 In the future, the results of this work will allow to determine value change patterns of one of four superposition coefficients of given four node points (both adjacent and non-adjacent) of different two-dimensional numerical sequences. This will make possible to solve problems of continuous discrete interpolation and extrapolation of any two-dimensional functional dependences by numerical sequences (such as determination the applicates of required points of discrete frames of two-dimensional geometric image) without laborious operations of compilation and solving huge systems of linear and transcendental equations.