Determination of absolute error of interpolation of curve with given geometric characteristics

Abstract

Purpose. The purpose of the research is to develop a method for determining of absolute error of interpolation of a point set by a non-oscillating plane curve from the condition of monotonous change of differential-geometric characteristics along the initial geometric image. Methodology. Analysis of methods of estimating of accuracy of interpolation by the methods of continuous geometric modeling showed that disadvantages of these methods are caused by the decision to compare the obtained model with a known predefined function, which differs from the initial curve to which the points set belongs. The method of determining the accuracy of the interpolation, which is based on the formation of curve on the basis of its geometrical characteristics, is proposed in this article. The assumption, on the basis of which the curve is formed, is following: if there is a curve that interpolates the original points set and there are no singular points for this curve (an inflection point, points of change direction of curvature, torsion values, etc.), then there are no such singular points on the original object. Two components of the occurrence of error are considered. The error, with which the formed curve that interpolates the original points set represents the original curve, is estimated as the area of possible location of all curves whose characteristics are identical to those of the original curve. The interpolating curve is forming in the form of a condensed points set, which consists of an arbitrarily large number of points determined from the condition of the possibility of interpolating its curve by a line with given characteristics. The error of the formation of the interpolating curve is estimated as the area of possible location of the curve interpolating the condensed points set. Findings. The solution of the problem for a plane curve on the basis of the condition of absence of oscillations and the condition of monotonous change of curvature values is present in this article. The area of location of curve, which is determined by the condition of convexity of curve, is maximal and is the initial. Addition of the following conditions (monotonous change of curvature along the curve and assignment of fixed positions of tangents and curvature values at initial points) localizes the area of possible solution. Originality. The developed method for estimating of accuracy of interpolation of a curve makes it possible to determine the absolute error with which the model represents the initial curve and the accuracy with which the interpolating curve represents any curve with given properties. Practical value. The developed method can be used in solving of problems that require the determination of maximum absolute error with which the model represents the original object. It is approximate calculations, the construction of graphics that describe of processes and phenomena, the formation of models of surfaces on the physical sample.