POSITIVITY PRESERVING INTERPOLATION BY USING RATIONAL CUBIC BALL SPLINE

Authors

  • Samsul Ariffin Abdul Karim Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia

DOI:

https://doi.org/10.11113/jt.v78.8107

Keywords:

Rational cubic Ball interpolant, positivity, dependent, visualization

Abstract

This paper discusses the positivity preserving by using rational cubic Ball interpolant of the form cubic/quadratic with two parameters. The sufficient condition for the positivity is derived on one parameter meanwhile the other one is a free parameter to control the final shape of the interpolating curves. The degree smoothness achieved is . From numerical results, the rational cubic Ball spline with two parameters gives smooth interpolating positive curves as well as visually pleasing for computer graphics visualization. Furthermore the scheme is better than existing schemes i.e. its easiness to use and less computation. All numerical results are produced by using Mathematica. 

References

Abbas, M., Majid, A. A., Awang, M. N. Hj. and Ali, J. M. 2013. Positivity-preserving Rational Cubic Spline Interpolation. Science Asia. 39: 208-213.

Brodlie, K. W. and Butt, S. 1991. Preserving Convexity Using Piecewise Cubic Interpolation. Computers and Graphics. 15: 15-23.

Brodlie, K. W., Mashwama, P. and Butt, S. 1995. Visualization Of Surface Data To Preserve Positivity And Other Simple Constraints. Computers and Graphics. 19(4): 585-594.

Brodlie, K. W., Asim, M.R. and Unsworth, K. 2005. Constrained Visualization Using Shepard Interpolation Family. Computers and Graphics Forum. 24(4): 809-820.

Butt, S. and Brodlie, K. W. 1993. Preserving Positivity Using Piecewise Cubic Interpolation. Computer and Graphics. 17(1): 55-64.

Goodman, T. N. T., Ong, B.H. and Unsworth, K. 1991. Constrained interpolation using rational cubic splines. NURBS for Curve and Surface Design. G. Farin (Ed.), SIAM, Philadelphia, 59-74.

Greiner, G. 1991. A survey on univariate data interpolation and approximation by splines of given shape. Mathematical Computer and Modeling. 15(10): 97-106.

Hussain, M.Z., Sarfraz, M. and Shaikh, T.S., 2011. Shape preserving rational cubic spline for positive and convex data. Egyptian Informatics Journal. 12: 231-236.

Hussain, M.Z. and Sarfraz, M. 2008. Positivity-Preserving interpolation of positive data by rational cubics, Journal of Computational and Applied Mathematics. 218: 446-458.

Hussain, M.Z., Hussain, M., Waseem, A. 2014. Shape-preserving trigonometric functions. Computational and Applied Mathematics. 33: 411–431.

Hussain, M.Z., Hussain, M. and Aqeel, B. 2014. Shape-preserving surfaces with constraints on tension parameters. Applied Mathematics and Computation, 247: 442-464.

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

Jaafar, W.N.W., Piah, A.R.Mt. and Abbas, M. 2015. Data visualization for positive data using rational cubic Ball function. AIP Conf. Proc. 1682. 020033-1- 020033-7.

Karim, S.A.A. 2013. Rational Cubic Ball Functions for Positivity Preserving. Far East Journal of Mathematical Sciences (FJMS). Vol. 82, No. 2, pp. 193-207.

Karim, S.A.A. 2014. Monotonic Interpolating Curves by Using Rational Cubic Ball Interpolation. Applied Mathematical Sciences, vol. 8, no. 146, 7259 – 7276.

Karim, S.A.A. 2015. Positivity preserving by using rational cubic Ball function. AIP Conf. Proc. 1660. 050049-1 - 050049-8.

Karim, S.A.A. 2015. Shape Preserving by Using Rational Cubic Ball Interpolant. Far East Journal of Mathematical Sciences (FJMS). 96(2): 211-230.

M. Sarfraz. 2012. Visualization of positive and convex data by a rational cubic spline interpolation,†Information Sciences. 146(1-4): 239-254.

Sarfraz, M., Hussain, M.Z. and Nisar. 2010. A. Positive data modeling using spline function. Applied Mathematics and Computation. 216: 2036-2049.

Sarfraz, M., Hussain, M.Z. and M Hussain. 2012. Shape-preserving curve interpolation, International Journal of Computer Mathematic. 89(1): 35-53.

Sarfraz, M., Hussain, M.Z. and Hussain, M. 2013. Modeling rational spline for visualization of shaped data. Journal of Numerical Mathematics. 21(1): 63-87.

Schmidt, J.W. and Hess, W. 1988. Positivity of cubic polynomials on intervals and positive spline interpolation. BIT. 28: 340-352.

Tahat, A.N.H., Piah, A.R.Mt. and Yahya, Z.R. 2015. Positivity preserving curves using rational cubic Ball interpolant. AIP Conf. Proc. 1682. 020016-1-020016-5.

Walther, M.B.- and Schmidt, J.W. 1999. Range Restricted Interpolation using Gregory’s Rational Cubic Splines. Journal of Computational and Applied Mathematics. 103: 221-237.

Wu, J., Zhang, X. and Peng, L. 2010. Positive Approximation and Interpolation Using Compactly Supported Radial Basis Functions. Mathematical Problems in Engineering. Article ID 964528, 10 pages.

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Published

2016-10-31

Issue

Section

Science and Engineering

How to Cite

POSITIVITY PRESERVING INTERPOLATION BY USING RATIONAL CUBIC BALL SPLINE. (2016). Jurnal Teknologi, 78(11). https://doi.org/10.11113/jt.v78.8107