DEVELOPMENT OF AN ALGORITHM FOR FORMING SPATIAL MONOTONE CURVES
Abstract
The formation of complex functional surfaces given by an array of points is an urgent task of geometric modeling.The geometric model of the surface is formed on the basis of a discrete linear framework, the elements of which are curved lines that interpolate sequences of points selected from the original array. Controlling the presence of singular points in the framework lines is an important condition for ensuring the necessary surface properties. The work solves the problem of modeling a smooth contour of the interpolating point series belonging to the curve line that contains the minimum number of singular points under the conditions of the problem. The solution of the problem of controlling the presence of singular points in the curve that interpolates the given point series is based on the rejection of its analytical representation. The interpolating curve is formed in the form of a region of possible location of its monotonic parts. All curved lines that meet the conditions of the problem cannot be located outside the specified region. As a linear element of the frame, a contour is used, which is formed inside the region of possible location of parts of the interpolating curve, along which the curvature values monotonically increase or decrease. The absolute error of interpolation of the point series is estimated by the width of the specified region. If the interpolation error of the original point series turns out to be greater than the assigned value, then the width of the region of the curve location is reduced by assigning intermediate points. The algorithm proposed in the work for modeling spatial linear elements of the frame while controlling the occurrence of special points is based on the formation of auxiliary flat discretely represented curves, the characteristics of which determine the configuration and properties of spatial contours. Based on the proposed algorithm, software has been developed for the automated formation in the CAD system of contours representing curved lines from the surface determinant.
References
2. Hashemian A., Imani B. A new quality appearance evaluation technique for automotive bodies including effect of flexible parts tolerances. Mechanics Based Design of Structures and Machines. 2017. Vol. 46, no. 2. P. 157–167. https://doi.org/10.1080/15397734.2017.1321487.
3. Fooladi M., Foroud A. Recognition and assessment of different factors which affect flicker in wind turbine. IET Renewable Power Generation. 2016. Vol. 10, no. 2. P. 250–259. https://doi. org/10.1049/iet-rpg.2014.0419.
4. Farhad Hosseini S., Moetakef-Imani B. Innovative approach to computer-aided design of horizontal axis wind turbine blades. Journal of Computational Design and Engineering. 2017. Vol. 4. Iss. 2. P. 98–105. https://doi.org/10.1016/j.jcde.2016.11.001.
5. Qawaqzeh M., Szafraniec A., Halko S., Miroshnyk O., Zharkov A. Modelling of a household electricity supply system based on a wind power plant. Przegląd Elektrotechniczny. 2020. Vol. 96. https://doi.org/10.15199/48.2020.11.08.
6. Havrylenko Y., Kholodniak Y., Halko S., Vershkov O., Bondarenko L., Suprun O., Miroshnyk O., Shchur T., Śrutek M., Gackowska M. Interpolation with Specified Error of a Point Series Belonging to a Monotone Curve. Entropy. 2021. Vol. 23, no. 5. P. 493. https://doi.org/10.3390/e23050493.
7. Ivan M. A note on the Hermite interpolation. Numerical Algorithms. 2014. Vol. 69, no. 3. P. 517–522. https://doi.org/10.1007/s11075-014-9909-x.
8. Argyros I.K., George S. On the convergence of Newton-like methods using restricted domains. Numerical Algorithms. 2016. Vol. 75, no. 3. P. 553–567. URL: https://doi.org/10.1007/ s11075-016-0211-y.
9. Samreen S., Sarfraz M., Hussain M. Z. A quadratic trigonometric spline for curve modeling. PLOS ONE. 2019. Vol. 14, no. 1. P. e0208015. URL: https://doi.org/10.1371/journal.pone.0208015.
10. Li H., Zhang L. Geometric error control in the parabola-blending linear interpolator. Journal of Systems Science and Complexity. 2013. Vol. 26, no. 5. P. 777–798. https://doi.org/10.1007/ s11424-013-3178-y.
11. Shen W., Wang G., Huang F. Direction monotonicity for a rational Bézier curve. Applied Mathematics-A Journal of Chinese Universities. 2016. Vol. 31, no. 1. P. 1–20. URL: https://doi. org/10.1007/s11766-016-3399-7.
12. Volkov Y.S. Obtaining a banded system of equations in complete spline interpolation problem via B-spline basis. Central European Journal of Mathematics. 2011. Vol. 10, no. 1. P. 352–356. https://doi.org/10.2478/s11533-011-0104-1.
13. Weishi Li, Shuhong Xu, Jianmin Zheng, Gang Zhao. Target curvature driven fairing algorithm for planar cubic B-spline curves. Computer Aided Geometric Design. 2004. Vol. 21. Iss. 5. P. 499–513. https://doi.org/10.1016/j.cagd.2004.03.004.
14. Hongli W., Wei Z. Modeling of the no-tillage planter and simulation of the cutting-stubble knife. System Science, Engineering Design and Manufacturing Informatization. 2012. P. 335–338. https://doi.org/10.1109/ICSSEM.2012.6340880.
15. Гавриленко Є. А., Холодняк Ю. В, Мірошниченко М. Ю. Алгоритм моделювання одновимірних обводів за заданими умовами. Науковий вісник Таврійського державного агротехнологічного університету. 2023. Вип. 12, т. 1. С. 1–9. https://doi.org/10.31388/sbtsatu.v12i1.301.
16. Холодняк Ю. В. Варіативне дискретне геометричне моделювання обводів на основі базисних трикутників по заданому закону зміни кривини : автореф. дис. … канд. техн. наук. Мелітополь, 2016. 24 с.
