DISCRETE INTERPOLATION BY SUPERPOSITIONS OF COORDINATES OF THREE POINTS OF ONE-DIMENSIONAL NUMERICAL SEQUENCES USING LINEAR FRACTIONAL FUNCTIONS

Abstract

A geometric image of an arbitrary shape can always be represented by an ordered set of points according to a certain law so that it becomes possible to determine coordinates of any point inside the contour (domain). We have got only a problem of a necessary density of initial information and costs of its obtaining, processing and storage.

For an adequate presentation of information about an object of study, it is necessary to organize processing and storage of significant amounts of information. This involves usage of powerful computers with a large volume of hard disk and RAM.

Usually in geometric modeling, initial data are geometric characteristics (coordinates or parameter values) and conditions, which represented in a numerical form and can be quite large. Because of this, global continuous modeling methods of searching a single solution are ineffective. It happens because they demand usage of sufficiently complex mathematical algorithms and cannot provide a necessary adequacy of the models. Discrete geometric modeling methods don’t have such disadvantages.

In this article we propose using the geometric apparatus of superpositions in combination with the classical finite difference method. It can significantly increase efficiency and expand capabilities of a discrete geometric modeling process. In particular, this allows us to investigate the possibility of using both parabolic and any other functional dependencies as interpolants.

Based on the geometric apparatus of superpositions, general formulas are obtained for calculating superposition coefficients of three given arbitrary points of one-dimensional numerical sequences. The sequences represent infinite discrete forms of certain functional dependencies. The found coefficients are used for calculating coordinates of unknown nodal points of these sequences.

Using the linear-fractional function, it is shown that obtained formulas for calculating superposition coefficients of given three nodal points for selected computational schemes allow us to solve problems of continuous discrete interpolation and extrapolation by numerical sequences of any one-dimensional functional dependencies, such as determining the ordinates of any desired points of the discrete curves by three given ordinates of nodal points, without complicate operation of compiling and solving large systems of linear equations.

Keywords: discrete modeling, geometric images, finite difference method, geometric apparatus of superpositions, linear fractional functions.