Shape Preserving C2 Rational Cubic Spline Interpolation


  • Abdurkadir Edeo Bule Hora College
  • Genanew Gonfa
  • Teshome Tefera


Shape preservation, Spline interpolation, Positivity, Monotonicity, Convexity


In this study a piecewise rational function  with cubic numerator and linear denominator involving two shape parameters has been developed to address the problem of constructing positivity preserving curve through positive data, monotonicity preserving curve through monotone data and convexity preserving curve through convex data within one mathematical model. A simple data dependent condition for a single shape parameter has been derived to preserve the positivity, monotonicity and convexity of respectively positive, monotone and convex data. The remaining shape parameter is left free for the user to modify the shape of positive, monotone and convex curves when needs arise. We extended the result of [1] to a piecewise rational cubic function.

Author Biography

Abdurkadir Edeo, Bule Hora College

Mathematics, Msc


[1] M. Sarfraz, M.Z. Hussain & M. Hussain. "Shape preserving curve interpolation." International Journal of Computer Mathematics, vol.89, pp.35-53, 2012.
[2] M.Z. Hussain, M. Sarfraz and T.S. Shaikh. "Shape preserving rational cubic spline for positive and convex data." Egyptian Informatics Journal,vol. 12, pp.231–236, 2011.
[3] M.Abbas, A.Abd.Majid , Hj. Awang, Md.N. and J. Md. Ali. "Positivity-preserving C2 rational cubic spline Interpolation." ScienceAsia, vol.39,pp.208–213,2012.
[4] G. Beliakov. "Monotonicity Preserving Approximation Of Multivariate Scattered Data." BIT Numer. Math., vol.45, pp. 653–677,Aug.2005.
[5] M.Z. Hussain and M.Sarfraz. "Positivity-preserving interpolation of positive data by rational cubics." J. Comput. Appl. Math,. vol.218, pp. 446–458,2008.
[6] M.Abbas, A.Abd. Majid, and J. Md. Ali. "A Rational Spline for Preserving the Shape of Positive Data." International Journal of Computer and Electrical Engineering, vol.5,2013.
[7] M.Abbas, A.Abd. Majid, and J. Md. Ali. "Monotonicity-preserving C2 rational cubic spline for monotone data." Applied Mathematics and Computation, vol.219, pp.2885–2895 ,2012.
[8] M.Abbas, A. Abd. Majid, Hj. Awang, Md.N. and J.Md. Ali . "Monotonicity Preserving Interpolation using Rational Spline." Proceedings of the International multi Conference of Engineers and Computer scientists I, 2011.
[9] M.Tian. "Monotonicity-Preserving Piecewise Rational Cubic Interpolation." Int. Journal of Math. Analysis, vol.5, pp. 99 – 104,2011.
[10] T.N.T. Goodman. "Shape preserving interpolation by curves." In Algorithms for Approximation IV Proceedings, 2002, pp. 24–35.
[11] P.Lamberti and C.Manni. "Shape-preserving C2 functional interpolation via parametric cubics." Numer. Algorithms, vol.28, pp. 229–254,2001.
[12] J.C.Fiorot and J.Tabka. "Shape-Preserving C2 Cubic Polynomial Interpolating Splines." Mathematics of Computation, vol.57, Pages 291-298,1991.
[13] M.Shrivastava and J.Joseph. "C2-rational cubic spline involving tension parameters." India Proc. Indian Acad. Sci. (Math. Sci.), Vol. 110, pp. 305-314, Aug.2000.




How to Cite

Edeo, A., Gonfa, G., & Tefera, T. (2015). Shape Preserving C2 Rational Cubic Spline Interpolation. American Scientific Research Journal for Engineering, Technology, and Sciences, 12(1), 110–122. Retrieved from