# MODELING OF ASYMPTOTICALLY OPTIMAL PIECEWISE LINEAR INTERPOLATION OF PLANE PARAMETRIC CURVES

## Authors

• О. V. Frolov Simon Kuznets Kharkiv National University Of Economic, Kharkiv, Ukraine., Ukraine
• M. U. Losev Simon Kuznets Kharkiv National University Of Economic, Kharkiv, Ukraine., Ukraine

## Keywords:

interpolation, polyline segment, linear rational B-spline, equidistant, integration, parametric curve, approximation error, variance.

## Abstract

Context. Piecewise linear approximation of curves has a large number of applications in computer algorithms, as the reconstruction of objects of complex shapes on monitors, CNC machines and 3D printers. In many cases, it is required to have the smallest number of segments for a given accuracy.

Objective. The objective of this paper is to improve the method of asymptotically optimal piecewise linear interpolation of plane parametric curves. This improvement is based to research influence of the method parameters and algorithms to distributions of approximation errors.

Method. An asymptotically optimal method of curves interpolation is satisfied to the condition of minimum number of approximation units. Algorithms for obtaining the values of the sequence of approximation nodes are suggested. This algorithm is based on numerical integration of the nodes regulator function with linear and spline interpolation of its values. The method of estimating the results of the curve approximation based on statistical processing of line segments sequence of relative errors is substantiated. Modeling of real curves approximation is carried out and influence of the sampling degree of integral function – the nodes regulator on distribution parameters of errors is studied. The influence is depending on a method of integral function interpolation.

Results. Research allows to define necessary the number of discretization nodes of the integral function in practical applications. There have been established that with enough sampling points the variance of the error’s distribution stabilizes and further increasing this number does not significantly increase the accuracy of the curve approximation. In the case of spline interpolation of the integral function, the values of the distribution parameters stabilized much faster, which allows to reduce the number of initial sampling nodes by 5–6 times having similar accuracy.

Conclusions. Modelling of convex planar parametric curves reconstruction by an asymptotically optimal linear interpolation algorithm showed acceptable results without exceeding the maximum errors limit in cases of a sufficient discretization of the integral function. The prospect of further research is to reduce the computational complexity when calculating the values of the integral distribution function by numerical methods, and to use discrete analogues of derivatives in the expression of this function.

## Author Biographies

### О. V. Frolov, Simon Kuznets Kharkiv National University Of Economic, Kharkiv, Ukraine.

PhD, Associate Professor, Associate Professor at the Information Systems Department.

### M. U. Losev, Simon Kuznets Kharkiv National University Of Economic, Kharkiv, Ukraine.

PhD, Associate Professor, Associate Professor at the Information Systems Department.

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2021-10-06

## How to Cite

Frolov О. V., & Losev, M. U. (2021). MODELING OF ASYMPTOTICALLY OPTIMAL PIECEWISE LINEAR INTERPOLATION OF PLANE PARAMETRIC CURVES . Radio Electronics, Computer Science, Control, (3), 57–68. https://doi.org/10.15588/1607-3274-2021-3-6

## Section

Mathematical and computer modelling