`ISRN Mathematical AnalysisVolume 2012 (2012), Article ID 174048, 14 pageshttp://dx.doi.org/10.5402/2012/174048`
Research Article

## Local Convexity Shape-Preserving Data Visualization by Spline Function

1School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia
2Department of Mathematics, University of Sargodha, 40100 Sargodha, Pakistan
3School of Distance Education, Universiti Sains Malaysia, 11800 Penang, Malaysia

Received 16 November 2011; Accepted 21 December 2011

Academic Editor: R. Barrio

Copyright © 2012 Muhammad Abbas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. F. Bao, Q. Sun, J. Pan, and Q. Duan, “Point control of rational interpolating curves using parameters,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 143–151, 2010.
2. M. Abbas, A. A. Majid, M. N. H. Awang, and J. M. Ali, “Monotonicity preserving interpolation using rational spline,” in Proceedings of the International MultiConference of Engineers and Computer Scientists (IMECS '11), vol. 1, pp. 278–282, Hong Kong, March 2011.
3. S. Asaturyan, P. Costantini, and C. Manni, “Local shape-preserving interpolation by space curves,” IMA Journal of Numerical Analysis, vol. 21, no. 1, pp. 301–325, 2001.
4. K. W. Brodlie and S. Butt, “Preserving convexity using piecewise cubic interpolation,” Computers and Graphics, vol. 15, no. 1, pp. 15–23, 1991.
5. J. M. Carnicer, M. Garcia-Esnaola, and J. M. Peña, “Convexity of rational curves and total positivity,” Journal of Computational and Applied Mathematics, vol. 71, no. 2, pp. 365–382, 1996.
6. J. C. Clements, “A convexity-preserving ${C}^{2}$ parametric rational cubic interpolation,” Numerische Mathematik, vol. 63, no. 2, pp. 165–171, 1992.
7. P. Costantini, “On monotone and convex spline interpolation,” Mathematics of Computation, vol. 46, no. 173, pp. 203–214, 1986.
8. P. Costantini and F. Fontanella, “Shape-preserving bivariate interpolation,” SIAM Journal on Numerical Analysis, vol. 27, no. 2, pp. 488–506, 1990.
9. 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.
10. J. A. Gregory, “Shape preserving spline interpolation,” Computer-Aided Design, vol. 18, no. 1, pp. 53–57, 1986.
11. M. Tian and S. L. Li, “Convexity-preserving piecewise rational cubic interpolation,” Journal of Shandong University, vol. 42, no. 10, pp. 1–5, 2007.
12. D. F. McAllister and J. A. Roulier, “An algorithm for computing a shape-preserving osculatory quadratic spline,” ACM Transactions on Mathematical Software, vol. 7, no. 3, pp. 331–347, 1981.
13. E. Passow and J. A. Roulier, “Monotone and convex spline interpolation,” SIAM Journal on Numerical Analysis, vol. 14, no. 5, pp. 904–909, 1977.
14. J. A. Roulier, “A convexity preserving grid refinement algorithm for interpolation of bivariate functions,” IEEE Computer Graphics and Applications, vol. 7, no. 1, pp. 57–62, 1987.
15. L. L. Schumaker, “On shape preserving quadratic spline interpolation,” SIAM Journal on Numerical Analysis, vol. 20, no. 4, pp. 854–864, 1983.
16. M. H. Schultz, Spline Analysis, Prentice-Hall, Englewood Cliffs, NJ, USA, 1973.
17. 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.
18. 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.
19. M. Sarfraz, M. Hussain, and Z. Habib, “Local convexity preserving rational cubic spline curves,” in Proceedings of the IEEE Conference on Information Visualization (IV '97), pp. 211–218, 1997.
20. M. Sarfraz, “Convexity preserving piecewise rational interpolation for planar curves,” Bulletin of the Korean Mathematical Society, vol. 29, no. 2, pp. 193–200, 1992.
21. M. Sarfraz, “Interpolatory rational cubic spline with biased, point and interval tension,” Computers and Graphics, vol. 16, no. 4, pp. 427–430, 1992.