Bézier curves are essential for data interpolation. However, traditional Bézier curves often fail to detect special features that may exist in a data set, such as monotonicity or convexity, leading to invalid interpolations. This study aims to improve the deficiency of Bézier curves by imposing monotonicity or convexity-preserving conditions on the shape parameter and control points. For this purpose, the quintic trigonometric Bézier curves with two shape parameters are used. These techniques constrain only one of the shape parameters, leaving the other free to provide users with more freedom and flexibility in modifying the final curve. To guarantee smooth interpolation, the curvature profiles of the curves are analyzed, which aids in selecting the optimal shape parameter values. The effectiveness of the developed schemes was evaluated by implementing real-life data and data obtained from the existing schemes. Compared with the existing schemes, the developed schemes produce low-curvature interpolation curves with unnoticeable wiggles and turns. The proposed methods also work effectively for both nonuniformly spaced data and negative-valued convex data in real-life applications. When the shape parameter is correctly chosen, the developed interpolants exhibit continuous curvature plots, assuring $ C^2 $ continuity.
Citation: Salwa Syazwani Mahzir, Md Yushalify Misro, Kenjiro T. Miura. Preserving monotone or convex data using quintic trigonometric Bézier curves[J]. AIMS Mathematics, 2024, 9(3): 5971-5994. doi: 10.3934/math.2024292
Bézier curves are essential for data interpolation. However, traditional Bézier curves often fail to detect special features that may exist in a data set, such as monotonicity or convexity, leading to invalid interpolations. This study aims to improve the deficiency of Bézier curves by imposing monotonicity or convexity-preserving conditions on the shape parameter and control points. For this purpose, the quintic trigonometric Bézier curves with two shape parameters are used. These techniques constrain only one of the shape parameters, leaving the other free to provide users with more freedom and flexibility in modifying the final curve. To guarantee smooth interpolation, the curvature profiles of the curves are analyzed, which aids in selecting the optimal shape parameter values. The effectiveness of the developed schemes was evaluated by implementing real-life data and data obtained from the existing schemes. Compared with the existing schemes, the developed schemes produce low-curvature interpolation curves with unnoticeable wiggles and turns. The proposed methods also work effectively for both nonuniformly spaced data and negative-valued convex data in real-life applications. When the shape parameter is correctly chosen, the developed interpolants exhibit continuous curvature plots, assuring $ C^2 $ continuity.
| [1] |
M. Z. Hussain, M. Hussain, Visualization of data preserving monotonicity, Appl. Math. Comput., 190 (2007), 1353–1364. http://dx.doi.org/10.1016/j.amc.2007.02.022 doi: 10.1016/j.amc.2007.02.022
|
| [2] |
M. Sarfraz, M. Z. Hussain, M. Hussain, Shape-preserving curve interpolation, Int. J. Comput. Math., 89 (2012), 35–53. http://dx.doi.org/10.1080/00207160.2011.627434 doi: 10.1080/00207160.2011.627434
|
| [3] |
M. Hussain, A. Abd Majid, M. Z. Hussain, Convexity-preserving bernstein-bézier quartic scheme, Egypt. Inform. J., 15 (2014), 89–95. http://dx.doi.org/10.1016/j.eij.2014.04.001 doi: 10.1016/j.eij.2014.04.001
|
| [4] |
B. Kvasov, Monotone and convex interpolation by weighted quadratic splines, Adv. Computat. Math., 40 (2014), 91–116. http://dx.doi.org/10.1007/s10444-013-9300-9 doi: 10.1007/s10444-013-9300-9
|
| [5] |
S. Karim, K. Pang, Monotonicity preserving using gc 1 rational quartic spline, AIP Conf. Proc., 1482 (2012), 26–31. http://dx.doi.org/10.1063/1.4757432 doi: 10.1063/1.4757432
|
| [6] | A. Edeo, G. Gofeb, T. Tefera, Shape preserving $C^{2}$ rational cubic spline interpolation, ASRJETS, 12 (2015), 110–122. |
| [7] |
S. Karim, Rational cubic spline interpolation for monotonic interpolating curve with $C^{2}$ continuity, MATEC Web Conf., 131 (2017), 04016. http://dx.doi.org/10.1051/matecconf/201713104016 doi: 10.1051/matecconf/201713104016
|
| [8] | A. Ahmad, M. Misro, Preserving monotonicity of ball curve and it's curvature profile, Proceedings of 6th IEEE International Conference on Recent Advances and Innovations in Engineering (ICRAIE), 2021, 1–6. http://dx.doi.org/10.1109/ICRAIE52900.2021.9704025 |
| [9] |
A. Tahat, A. Piah, Z. Yahya, Rational cubic ball curves for monotone data, AIP Conf. Proc., 1750 (2016), 030021. http://dx.doi.org/10.1063/1.4954557 doi: 10.1063/1.4954557
|
| [10] | A. Ahmad, M. Misro, Curvature comparison of bézier curve, ball curve and trigonometric curve in preserving the positivity of real data, AMCI, 11 (2022), 12–20. |
| [11] |
F. Pitolli, Ternary shape-preserving subdivision schemes, Math. Comput. Simulat., 106 (2014), 185–194. http://dx.doi.org/10.1016/j.matcom.2013.04.003 doi: 10.1016/j.matcom.2013.04.003
|
| [12] |
P. Ashraf, M. Sabir, A. Ghaffar, K. Nisar, I. Khan, Shape-preservation of the four-point ternary interpolating non-stationary subdivision scheme, Front. Phys., 7 (2020), 241. http://dx.doi.org/10.3389/fphy.2019.00241 doi: 10.3389/fphy.2019.00241
|
| [13] |
A. Chand, N. Vijender, M. Navascués, Shape preservation of scientific data through rational fractal splines, Calcolo, 51 (2014), 329–362. http://dx.doi.org/10.1007/s10092-013-0088-2 doi: 10.1007/s10092-013-0088-2
|
| [14] |
P. Viswanathan, A. Chand, A fractal procedure for monotonicity preserving interpolation, Appl. Math. Comput., 247 (2014), 190–204. http://dx.doi.org/10.1016/j.amc.2014.06.090 doi: 10.1016/j.amc.2014.06.090
|
| [15] |
L. Peng, Y. Zhu, $C^{1}$ convexity-preserving piecewise variable degree rational interpolation spline, J. Adv. Mech. Des. Syst., 14 (2020), JAMDSM0002. http://dx.doi.org/10.1299/jamdsm.2020jamdsm0002 doi: 10.1299/jamdsm.2020jamdsm0002
|
| [16] |
M. Sarfraz, Visualization of positive and convex data by a rational cubic spline interpolation, Inform. Sciences, 146 (2002), 239–254. http://dx.doi.org/10.1016/S0020-0255(02)00209-8 doi: 10.1016/S0020-0255(02)00209-8
|
| [17] |
M. Abbas, A. Abd Majid, J. Ali, Local convexity-preserving rational cubic spline for convex data, Sci. World J., 2014 (2014), 391568. http://dx.doi.org/10.1155/2014/391568 doi: 10.1155/2014/391568
|
| [18] |
Y. Zhu, $C^{2}$ rational quartic/cubic spline interpolant with shape constraints, Results Math., 73 (2018), 127. http://dx.doi.org/10.1007/s00025-018-0883-9 doi: 10.1007/s00025-018-0883-9
|
| [19] |
M. Z. Hussain, M. Hussain, A. Waseem, Shape-preserving trigonometric functions, Comp. Appl. Math., 33 (2014), 411–431. http://dx.doi.org/10.1007/s40314-013-0071-1 doi: 10.1007/s40314-013-0071-1
|
| [20] |
V. Bogdanov, Y. Volkov, Near-optimal tension parameters in convexity preserving interpolation by generalized cubic splines, Numer. Algor., 86 (2021), 833–861. http://dx.doi.org/10.1007/s11075-020-00914-9 doi: 10.1007/s11075-020-00914-9
|
| [21] |
X. Han, Y. Ma, X. Huang, The cubic trigonometric bézier curve with two shape parameters, Appl. Math. Lett., 22 (2009), 226–231. http://dx.doi.org/10.1016/j.aml.2008.03.015 doi: 10.1016/j.aml.2008.03.015
|
| [22] |
S. Maqsood, M. Abbas, G. Hu, A. Ramli, K. Miura, A novel generalization of trigonometric bézier curve and surface with shape parameters and its applications, Math. Probl. Eng., 2020 (2020), 4036434. http://dx.doi.org/10.1155/2020/4036434 doi: 10.1155/2020/4036434
|
| [23] |
M. Misro, A. Ramli, J. Ali, Quintic trigonometric bézier curve with two shape parameters, Sains Malays., 46 (2017), 825–831. http://dx.doi.org/10.17576/jsm-2017-4605-17 doi: 10.17576/jsm-2017-4605-17
|
| [24] |
M. Misro, A. Ramli, J. Ali, Quintic trigonometric bézier curve and its maximum speed estimation on highway designs, AIP Conf. Proc., 1974 (2018), 020089. http://dx.doi.org/10.1063/1.5041620 doi: 10.1063/1.5041620
|
| [25] |
V. Bulut, Path planning for autonomous ground vehicles based on quintic trigonometric bézier curve: path planning based on quintic trigonometric bézier curve, J. Braz. Soc. Mech. Sci. Eng., 43 (2021), 104. http://dx.doi.org/10.1007/s40430-021-02826-8 doi: 10.1007/s40430-021-02826-8
|
| [26] |
J. Li, D. Zhao, An investigation on image compression using the trigonometric bézier curve with a shape parameter, Math. Probl. Eng., 2013 (2013), 731648. http://dx.doi.org/10.1155/2013/731648 doi: 10.1155/2013/731648
|
| [27] |
N. Ismail, M. Misro, Surface construction using continuous trigonometric bézier curve, AIP Conf. Proc., 2266 (2020), 040012. http://dx.doi.org/10.1063/5.0018101 doi: 10.1063/5.0018101
|
| [28] |
M. Z. Hussain, M. Hussain, Z. Yameen, A ${C}^{2}$-continuous rational quintic interpolation scheme for curve data with shape control, J. Nati. Sci. Found. Sri, 46 (2018), 341–354. http://dx.doi.org/10.4038/jnsfsr.v46i3.8486 doi: 10.4038/jnsfsr.v46i3.8486
|
| [29] |
S. Graiff Zurita, K. Kajiwara, K. Miura, Fairing of planar curves to log-aesthetic curves, Japan J. Indust. Appl. Math., 40 (2023), 1203–1219. http://dx.doi.org/10.1007/s13160-023-00567-w doi: 10.1007/s13160-023-00567-w
|
| [30] |
S. Mahzir, M. Misro, Shape preserving interpolation of positive and range-restricted data using quintic trigonometric bézier curves, Alex. Eng. J., 80 (2023), 122–133. http://dx.doi.org/10.1016/j.aej.2023.08.009 doi: 10.1016/j.aej.2023.08.009
|
Figures(9) / Tables(6)
Salwa Syazwani Mahzir, Md Yushalify Misro, Kenjiro T. Miura. Preserving monotone or convex data using quintic trigonometric Bézier curves[J]. AIMS Mathematics, 2024, 9(3): 5971-5994. doi: 10.3934/math.2024292
DownLoad: