ANALYSIS OF CONTINUOUS INTERPOLATION METHODS AND APPROXIMATION OF A FLAT CURVE

Abstract

The article provides an overview of continuous interpolation and approximation methods that are designed for discretely presented source data. It has been established that a common drawback is the lack of the possibility of local adjustment of the result of geometric modeling without re-computing the entire solution to the problem. To fulfill the conditions for the interpolation curve to pass through predetermined points, it is necessary to solve a system of linear equations. An increase in the size of the corresponding determinant leads to an increase in the error of the solution.

It should be noted that, depending on the initial information and the purpose of the research, methods of continuous or discrete interpolation and approximation can be used. This article analyzes directly continuous methods of discrete interpolation and approximation.

If we consider splines, then the parameters that control the shape of its individual segments are determined by the points that are the source data and determine the discretely presented curve. Spline methods are studied in the works of Yu.S. Zavyalova, N.P. Korneychuk and D. Rogers, of which it is known that the most important characteristics of a spline are its degree and defect. The degree of spline is determined by the highest degree of segment from the segments that make up the spline. A spline defect is the smallest derivative in which a discontinuity occurs, of all derivatives at the edges of segments that are spline components.

In applied geometry, second-order curves are most often used due to the developed geometric apparatus and the high adaptability of their application. The limitations for their use in the interpolation process is the obligatory arrangement of the source data in accordance with the shape defined by the second-order curve. Although, in the process of approximation, this restriction is less significant, but the reproduction of the curve should be performed with a certain, predetermined error.

Key words: approximation, interpolation, plane curves, Bezier curves, splines.