## Local convexity shape-preserving data visualization by spline function.(English)Zbl 1245.65018

Summary: The main purpose of this paper is the visualization of convex data that results in a smooth, pleasant, and interactive convexity-preserving curve. The rational cubic function with three free parameters is constructed to preserve the shape of convex data. The free parameters are arranged in a way that two of them are left free for user choice to refine the convex curve as desired, and the remaining one free parameter is constrained to preserve the convexity everywhere. Simple data-dependent constraints are derived on one free parameter, which guarantee to preserve the convexity of curve. Moreover, the scheme under discussion is, $$C^1$$ flexible, simple, local, and economical as compared to existing schemes. The error bound for the rational cubic function is $$O(h^3)$$.

### MSC:

 65D17 Computer-aided design (modeling of curves and surfaces) 65D07 Numerical computation using splines
Full Text:

### References:

 [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. · Zbl 1201.65021 [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. · Zbl 0976.65007 [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. · Zbl 0853.65019 [6] J. C. Clements, “A convexity-preserving C2 parametric rational cubic interpolation,” Numerische Mathematik, vol. 63, no. 2, pp. 165-171, 1992. · Zbl 0763.41001 [7] P. Costantini, “On monotone and convex spline interpolation,” Mathematics of Computation, vol. 46, no. 173, pp. 203-214, 1986. · Zbl 0617.41015 [8] P. Costantini and F. Fontanella, “Shape-preserving bivariate interpolation,” SIAM Journal on Numerical Analysis, vol. 27, no. 2, pp. 488-506, 1990. · Zbl 0707.41001 [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. · Zbl 0586.65006 [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. · Zbl 1174.65315 [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. · Zbl 0464.65003 [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. · Zbl 0378.41002 [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. · Zbl 0521.65009 [16] M. H. Schultz, Spline Analysis, Prentice-Hall, Englewood Cliffs, NJ, USA, 1973. · Zbl 0333.41009 [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. · Zbl 1086.65010 [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. · Zbl 1033.68681 [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. · Zbl 0763.65004 [21] M. Sarfraz, “Interpolatory rational cubic spline with biased, point and interval tension,” Computers and Graphics, vol. 16, no. 4, pp. 427-430, 1992.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.