Research Article | OPEN ACCESS
Shape Preserving Interpolation using Rational Cubic Spline
1Samsul Ariffin Abdul Karim and 2Kong Voon Pang
1Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 31750 Tronoh, Perak Darul Ridzuan, Malaysia
2School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Minden, Penang, Malaysia
Research Journal of Applied Sciences, Engineering and Technology 2014 2:167-178
Received: January 10, 2014 | Accepted: February 15, 2014 | Published: July 10, 2014
Abstract
This study proposes new C1 rational cubic spline interpolant of the form cubic/quadratic with three shape parameters to preserves the geometric properties of the given data sets. Sufficient conditions for the positivity and data constrained modeling of the rational interpolant are derived on one parameter while the remaining two parameters can further be utilized to change and modify the final shape of the curves. The sufficient conditions ensure the existence of positive and constrained rational interpolant. Several numerical results will be presented to test the capability of the proposed rational interpolant scheme. Comparisons with the existing scheme also have been done. From all numerical results, the new rational cubic spline interpolant gives satisfactory results.
Keywords:
Continuity, parameters, positivity preserving, rational cubic spline, shape preserving,
References
-
Abbas, M., A.A. Majid, M.N.Hj. Awang and J.M. Ali, 2012a. Shape preserving positive surface data visualization by spline functions. Appl. Math. Sci., 6(6): 291-307.
-
Abbas, M., A.A. Majid and J.M. Ali, 2012b. Monotonicity-preserving C2 rational cubic spline for monotone data. Appl. Math. Comput., 219: 2885-2895.
CrossRef
-
Abbas, M., A.A. Majid, M.N.Hj. Awang and J.M. Ali, 2013. Positivity-preserving C2 rational cubic spline interpolation. Sci. Asia, 39: 208-213.
CrossRef
-
Bashir, U. and J.M. Ali, 2013. Data visualization using rational trigonometric spline. J. Appl. Math., 2013: 10, Article ID 531497.
CrossRef
-
Brodlie, K.W. and S. Butt, 1991. Preserving convexity using piecewise cubic interpolation. Comput. Graph., 15: 15-23.
CrossRef
-
Butt, S. and K.W. Brodlie, 1993. Preserving positivity using piecewise cubic interpolation. Comput. Graph., 17(1): 55-64.
CrossRef
-
Delbourgo, R. and J.A. Gregory, 1985. The determination of derivative parameters for a monotonic rational quadratic interpolant. IMA J. Numer. Anal., 5: 397-406.
CrossRef
-
Dougherty, R.L., A. Edelman and J.M. Hyman, 1989. Nonnegativity-, monotonicity-, or convexity-preserving cubic and quintic hermite interpolation. Math. Comput., 52(186): 471-494.
CrossRef
-
Hussain, M.Z. and J.M. Ali, 2006. Positivity preserving piecewise rational cubic interpolation. Matematika, 22(2): 147-153.
-
Hussain, M.Z. and M. Hussain, 2006. Visualization of data subject to positive constraint. J. Inform. Comput. Sci., 1(3): 149-160.
-
Hussain, M.Z. and M. Sarfraz, 2008. Positivity-preserving interpolation of positive data by rational cubics. J. Comput. Appl. Math., 218: 446-458.
CrossRef
-
Hussain, M.Z., M. Sarfraz and T.S. Shaikh, 2011. Shape preserving rational cubic spline for positive and convex data. Egypt. Inform. J., 12: 231-236.
CrossRef
-
Ibraheem, F., M. Hussain, M.Z. Hussain and A.A. Bhatti, 2012. Positive data visualization using trigonometric function. J. Appl. Math., 2012: 19, Article ID 247120.
CrossRef
-
Sarfraz, M., 2002. Visualization of positive and convex data by a rational cubic spline interpolation. Inform. Sci., 146(1-4): 239-254.
CrossRef
-
Sarfraz, M., M.A. Mulhem and F. Ashraf, 1997. Preserving monotonic shape of the data using piecewise rational cubic functions. Comput. Graph., 21(1): 5-14.
CrossRef
-
Sarfraz, M., S. Butt and M.Z. Hussain, 2001. Visualization of shaped data by a rational cubic spline interpolation. Comput. Graph., 25: 833-845.
CrossRef
-
Sarfraz, M., M.Z. Hussain and A. Nisar, 2010. Positive data modeling using spline function. Appl. Math. Comput., 216: 2036-2049.
CrossRef
-
Sarfraz, M., M.Z. Hussain and F.S. Chaudary, 2005. Shape preserving cubic spline for data visualization. Comput. Graph. CAD/CAM, 01: 185-193.
-
Sarfraz, M., M.Z. Hussain and M. Hussain, 2013. Modeling rational spline for visualization of shaped data. J. Numer. Math., 21(1): 63-87.
CrossRef
-
Schmidt, J.W. and W. Hess, 1988. Positivity of cubic polynomials on intervals and positive spline interpolation. BIT, 28: 340-352.
CrossRef
-
Shaikh, T.S., M. Sarfraz and M.Z. Hussain, 2011. Shape preserving constrained data visualization using rational functions. J. Prime Res. Math., 7: 35-51.
-
Tian, M., Y. Zhang, J. Zhu and Q. Duan, 2005. Convexity-preserving piecewise rational cubic interpolation. J. Inform. Comput. Sci., 2(4): 799-803.
Competing interests
The authors have no competing interests.
Open Access Policy
This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Copyright
The authors have no competing interests.
|
|
![](http://www.maxwellsci.com/images/RJASET-maxw.jpg) |
ISSN (Online): 2040-7467
ISSN (Print): 2040-7459 |
![Submit Manuscript](../images/sub.jpg) |
Information |
|
|
|
Sales & Services |
|
|
|