## Journal of Applied Mathematics

### Shape Preserving Data Interpolation Using Rational Cubic Ball Functions

#### Abstract

A smooth curve interpolation scheme for positive, monotone, and convex data is developed. This scheme uses rational cubic Ball representation with four shape parameters in its description. Conditions of two shape parameters are derived in such a way that they preserve the shape of the data, whereas the other two parameters remain free to enable the user to modify the shape of the curve. The degree of smoothness is ${C}^{1}$. The outputs from a number of numerical experiments are presented.

#### Article information

Source
J. Appl. Math., Volume 2015 (2015), Article ID 908924, 9 pages.

Dates
First available in Project Euclid: 17 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1439816411

Digital Object Identifier
doi:10.1155/2015/908924

Mathematical Reviews number (MathSciNet)
MR3366494

#### Citation

Tahat, Ayser Nasir Hassan; Piah, Abd Rahni Mt; Yahya, Zainor Ridzuan. Shape Preserving Data Interpolation Using Rational Cubic Ball Functions. J. Appl. Math. 2015 (2015), Article ID 908924, 9 pages. doi:10.1155/2015/908924. https://projecteuclid.org/euclid.jam/1439816411

#### References

• M. Sarfraz and M. Z. Hussain, “Data visualization using rational spline interpolation,” Journal of Computational and Applied Mathematics, vol. 189, no. 1-2, pp. 513–525, 2006.
• M. Sarfraz, S. Butt, and M. Z. Hussain, “Visualization of shaped data by a rational cubic spline interpolation,” Computers & Graphics, vol. 25, no. 5, pp. 833–845, 2001.
• M. Z. Hussain, N. Ayub, and M. Irshad, “Visualization of 2D data by rational quadratic functions,” Journal of Information and Computing Science, vol. 2, no. 1, pp. 17–26, 2007.
• M. Sarfraz, M. Z. Hussain, and M. Hussain, “Shape-preserving curve interpolation,” International Journal of Computer Mathematics, vol. 89, no. 1, pp. 35–53, 2012.
• A. N. H. Tahat, A. R. M. Piah, and Z. R. Yahya, “Positivity preserving rational cubic Ball constrained interpolation,” in Proceedings of the 21st National Symposium on Mathematical Sciences (SKSM21 '13), pp. 331–336, AIP Publishing, Penang, Malaysia, November 2013.
• M. Z. Hussain and M. Hussain, “Visualization of data preserving monotonicity,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1353–1364, 2007.
• M. Z. Hussain and M. Sarfraz, “Monotone piecewise rational cubic interpolation,” International Journal of Computer Mathematics, vol. 86, no. 3, pp. 423–430, 2009.
• R. Delbourgo and J. A. Gregory, “Shape preserving piecewise rational interpolation,” SIAM Journal on Scientific and Statistical Computing, vol. 6, no. 4, pp. 967–976, 1985.
• M. Tian, “Monotonicity-preserving piecewise rational cubic interpolation,” International Journal of Mathematical Analysis, vol. 5, no. 1–4, pp. 99–104, 2011.
• A. R. M. Piah and K. Unsworth, “Improved sufficient conditions for monotonic piecewise rational quartic interpolation,” Sains Malaysiana, vol. 40, no. 10, pp. 1173–1178, 2011.
• Q. Wang and J. Tan, “Rational quartic spline involving shape parameters,” Journal of Information and Computational Science, vol. 1, no. 1, pp. 127–130, 2004.
• A. R. M. Piah and K. Unsworth, “Monotonicity preserving rational cubic ball interpolationčommentComment on ref. [12?]: Please update the information of this reference, if possible.,” In press.
• T. S. Shaikh, M. Sarfraz, and M. Z. Hussain, “Shape preserving positive and convex data visualization using rational bi-cubic functions,” Pakistan Journal of Statistics & Operation Research, vol. 8, no. 1, pp. 121–138, 2012.
• K. W. Brodlie and S. Butt, “Preserving convexity using piecewise cubic interpolation,” Computers & Graphics, vol. 15, no. 1, pp. 15–23, 1991.
• M. Sarfraz, “Convexity preserving piecewise rational interpolation for planar curves,” Bulletin of the Korean Mathematical Society, vol. 29, no. 2, pp. 193–200, 1992.
• M. Sarfraz, “Visualization of positive and convex data by a rational cubic spline interpolation,” Information Sciences, vol. 146, no. 1–4, pp. 239–254, 2002.
• M. Hussain, M. Z. Hussain, and M. Sarfraz, “Data visualization using spline functions,” Pakistan Journal of Statistics & Operation Research, vol. 9, no. 2, pp. 181–203, 2013.
• A. A. Ball, “CONSURF. Part one: introduction of the conic lofting tile,” Computer-Aided Design, vol. 6, no. 4, pp. 243–249, 1974.
• W. Guojin, “Ball curve of high degree and its geometric properties,” Applied Mathematics, vol. 2, no. 1, pp. 126–140, 1987.
• H. B. Said, “A generalized Ball curve and its recursive algorithm,” ACM Transactions on Graphics, vol. 8, no. 4, pp. 360–371, 1989.
• T. N. T. Goodman and H. B. Said, “Properties of generalized Ball curves and surfaces,” Computer-Aided Design, vol. 23, no. 8, pp. 554–560, 1991.
• T. N. T. Goodman and H. B. Said, “Shape preserving properties of the generalised Ball basis,” Computer Aided Geometric Design, vol. 8, no. 2, pp. 115–121, 1991. \endinput