Sains Malaysiana 48(3)(2019): 685–695

http://dx.doi.org/10.17576/jsm-2019-4803-23

 

Constrained Interpolation using Rational Cubic Spline with Three Parameters

(Interpolasi Berkekangan menggunakan Splin Kubus dengan Tiga Parameter)

 

SAMSUL ARIFFIN ABDUL KARIM¹, MOHAMMAD KHATIM HASAN² & ISHAK HASHIM^3*

 

¹Fundamental and Applied Sciences Department and Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia

 

²Pusat Penyelidikan Teknologi Kecerdasan Buatan, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia

 

³School of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia

 

Diserahkan: 27 Mac 2018/Diterima: 5 Disember 2018

 

ABSTRACT

The C¹ rational cubic spline function (cubic/quadratic) with three parameters is used to construct a constrained interpolating curve that lies below or above an arbitrary straight line or between two straight lines. The data dependent sufficient conditions for the rational cubic interpolant bounded by two straight lines are derived on one parameter, while the other two are free parameters that will be useful for shape modification. Some numerical results will be presented by using Mathematica software. Comparison with some existing schemes shows that the proposed scheme outperforms the existing schemes.

 

Keywords: Constrained interpolation; continuity; rational cubic spline

 

ABSTRAK

Fungsi splin kubus nisbah C¹ dengan tiga parameter digunakan untuk membentuk satu lengkung berkekangan yang berada di bawah atau atas suatu garis lurus atau di antara dua garisan lurus. Syarat cukup bagi lengkung kubus nisbah terbatas di antara dua garisan lurus yang bergantung kepada data dihasilkan pada satu parameter dengan dua yang lain adalah parameter bebas yang berguna untuk perubahan bentuk. Beberapa hasil berangka akan ditunjuk dengan menggunakan perisian Mathematica. Perbandingan dengan beberapa skema sedia ada menunjukkan bahawa skema yang dicadangkan mengatasi skema sedia ada.

 

Kata kunci: Interpolasi berkekangan; keselanjaran; splin kubus nisbah

RUJUKAN

Abbas, M., Majid, A.A., Awang, M.N.H. & Ali, J.M. 2012. Constrained shape preserving rational bi-cubic spline interpolation. World Applied Sciences Journal 20: 790-800.

Bashir, U. & Ali, J.M. 2013. Data visualization using rational trigonometric spline. Journal of Applied Mathematics 2013: 531497.

Bastian-Walther, M. & Schmidt, J.W. 1999. Range restricted interpolation using Gregory’s rational cubic splines. Journal of Computational and Applied Mathematics 103: 221-237.

Delbourgo, R. & Gregory, J.A. 1985. The determination of derivative parameters for a monotonic rational quadratic interpolant. IMA Journal of Numerical Analysis 5: 397-406.

Duan, Q., Wang, L. & Twizell, E.H. 2005. A new C² rational interpolation based on function values and constrained control of the interpolant curves. Applied Mathematics and Computation 161: 311-322.

Goodman, T.N.T., Ong, B.H. & Unsworth, K. 1991. Constrained interpolation using rational cubic splines. In NURBS for Curve and Surface Design (Geometric Design Publications), edited by Farin, G. Philadelphia: Society for Industrial & Applied. pp. 59-74.

Hussain, M.Z. & Hussain, M. 2006. Visualization of data subject to positive constraint. Journal of Information and Computing Sciences 1: 149-160.

Ibraheem, F., Hussain, M., Hussain, M.Z. & Bhatti, A.A. 2012. Positive data visualization using trigonometric polynomials. Journal of Applied Mathematics 2012: 247120.

Karim, S.A.A. & Kong, V.P. 2014. Shape preserving interpolation using rational cubic spline. Research Journal of Applied Sciences, Engineering and Technology 8: 167-168.

Meek, D.S., Ong, B.H. & Walton, D.J. 2003. Constrained interpolation with rational cubics. Computer Aided Geometric Design 20: 253-275.

Sarfraz, M., Hussain, M.Z. & Hussain, F. 2015. Shape preserving curves using quadratic trigonometric splines. Applied Mathematics and Computation 265: 1126-1144.

Sarfraz, M., Hussain, M.Z. & Hussain, M. 2013a. Modeling rational spline for visualization of shaped data. Journal of Numerical Mathematics 21: 63-87.

Sarfraz, M., Irshad, M. & Hussain, M.Z. 2013b. Reverse engineering of planar objects using GAs. Sains Malaysiana 42(8): 1167-1179.

Shaikh, T.S., Sarfraz, M. & Hussain, M.Z. 2011. Shape preserving constrained data visualization using rational functions. Journal of Prime Research in Mathematics 7: 35-51.

 

*Pengarang untuk surat-menyurat; email: ishak_h@ukm.edu.my