Development of the method for the formation of one-dimensional contours by the assigned interpolation accuracy
DOI:
https://doi.org/10.15587/1729-4061.2018.123921Keywords:
interpolation error, ordered set of points, oscillation, monotonic change of differential-geometric characteristicsAbstract
The purpose of the study is to develop a method for the formation of a one-dimensional contour with provision of a given accuracy of interpolation. Determination of the accuracy of interpolation relies on the formation of a curve based on known geometric properties. We build the geometric model with the assumption: if there is a curve line without special points that interpolates the point series, then there are no special points in the original object. Such points include: points of inflection, changes in the direction of growth along the curve of values of the curvature, the rounding, etc.
We build the interpolating curve in the form of a condensed point series consisting of arbitrarily large numbers of nodes, which are determined based on the possibility of interpolating their curve with a given line with the given characteristics. The error, with which the discrete representation of the curve represents the original curve, is evaluated as an area of the possible arrangement of all curves that interpolate an output point with properties identical to the properties of the original curve. We evaluate the error of formation of the interpolating curve line as the area of a possible location of the curve line that interpolates the condensed point series. In the study, we propose the solution of the problem for a flat curve based on the condition of the absence of oscillations and the conditions for the monotonous change of the curvature. The area of a location of the curve determined from the condition of the convexity of the curve is maximal and is the output one. Overlaying the following conditions: monotonous curvature change along the curve and the appointment of fixed positions of tangents and values of the curvature at the output points, localizes an area of a possible solution. One can use the developed method for solving problems requiring determination of the maximum absolute error with which a model represents the original object. These are approximate calculations, construction of graphs that describe processes and phenomena, formation of surface models representing existing physical samples.
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Copyright (c) 2018 Yevhen Havrylenko, Yuliia Kholodniak, Oleksandr Vershkov, Andrii Naidysh
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