Development of the method for the formation of one-dimensional contours by the assigned interpolation accuracy
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...
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doaj-7586e772feb94c19a6d5913ddbaf89e82020-11-25T02:19:31ZengPC Technology CenterEastern-European Journal of Enterprise Technologies1729-37741729-40612018-02-0114 (91)768210.15587/1729-4061.2018.123921123921Development of the method for the formation of one-dimensional contours by the assigned interpolation accuracyYevhen Havrylenko0Yuliia Kholodniak1Oleksandr Vershkov2Andrii Naidysh3Tavria State Agrotechnological University B. Khmelnytskoho аve., 18, Melitopol, Ukraine, 72310Tavria State Agrotechnological University B. Khmelnytskoho аve., 18, Melitopol, Ukraine, 72310Tavria State Agrotechnological University B. Khmelnytskoho аve., 18, Melitopol, Ukraine, 72310Bogdan Khmelnitsky Melitopol State Pedagogical University Hetmanska str., 20, Melitopol, Ukraine, 72312The 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.http://journals.uran.ua/eejet/article/view/123921interpolation errorordered set of pointsoscillationmonotonic change of differential-geometric characteristics |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Yevhen Havrylenko Yuliia Kholodniak Oleksandr Vershkov Andrii Naidysh |
spellingShingle |
Yevhen Havrylenko Yuliia Kholodniak Oleksandr Vershkov Andrii Naidysh Development of the method for the formation of one-dimensional contours by the assigned interpolation accuracy Eastern-European Journal of Enterprise Technologies interpolation error ordered set of points oscillation monotonic change of differential-geometric characteristics |
author_facet |
Yevhen Havrylenko Yuliia Kholodniak Oleksandr Vershkov Andrii Naidysh |
author_sort |
Yevhen Havrylenko |
title |
Development of the method for the formation of one-dimensional contours by the assigned interpolation accuracy |
title_short |
Development of the method for the formation of one-dimensional contours by the assigned interpolation accuracy |
title_full |
Development of the method for the formation of one-dimensional contours by the assigned interpolation accuracy |
title_fullStr |
Development of the method for the formation of one-dimensional contours by the assigned interpolation accuracy |
title_full_unstemmed |
Development of the method for the formation of one-dimensional contours by the assigned interpolation accuracy |
title_sort |
development of the method for the formation of one-dimensional contours by the assigned interpolation accuracy |
publisher |
PC Technology Center |
series |
Eastern-European Journal of Enterprise Technologies |
issn |
1729-3774 1729-4061 |
publishDate |
2018-02-01 |
description |
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. |
topic |
interpolation error ordered set of points oscillation monotonic change of differential-geometric characteristics |
url |
http://journals.uran.ua/eejet/article/view/123921 |
work_keys_str_mv |
AT yevhenhavrylenko developmentofthemethodfortheformationofonedimensionalcontoursbytheassignedinterpolationaccuracy AT yuliiakholodniak developmentofthemethodfortheformationofonedimensionalcontoursbytheassignedinterpolationaccuracy AT oleksandrvershkov developmentofthemethodfortheformationofonedimensionalcontoursbytheassignedinterpolationaccuracy AT andriinaidysh developmentofthemethodfortheformationofonedimensionalcontoursbytheassignedinterpolationaccuracy |
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1724876235527421952 |