Sains Malaysiana 45(10)(2016): 1557–1563

 

A Univariate Rational Quadratic Trigonometric Interpolating Spline to Visualize

Shaped Data

(Suatu Nisbah Kuadratik Trigonometri Univariat Menginterpolasikan Splin untuk Menggambarkan Data Berbentuk)

 

UZMA BASHIR¹* & JAMALUDIN MD. ALI²

 

¹Lahore College for Women University, Lahore, Pakistan

 

²School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Pulau Pinang

Malaysia

 

Received: 21 March 2015/Accepted: 25 February 2016

 

ABSTRACT

This study was concerned with shape preserving interpolation of 2D data. A piecewise C1 univariate rational quadratic trigonometric spline including three positive parameters was devised to produce a shaped interpolant for given shaped data. Positive and monotone curve interpolation schemes were presented to sustain the respective shape features of data. Each scheme was tested for plentiful shaped data sets to substantiate the assertion made in their construction. Moreover, these schemes were compared with conventional shape preserving rational quadratic splines to demonstrate the usefulness of their construction.

 

Keywords: Continuity; interpolation; rational trigonometric spline; shape preserving

 

ABSTRAK

Kajian ini adalah berkaitan dengan pemeliharaan bentuk interpolasi data 2D. Suatu cebisan splin nisbah kuadratik trigonometri univariat termasuk tiga parameter positif dirangka untuk menghasilkan interpolasi yang terbentuk daripada bentuk data yang diberikan. Skim lengkung interpolasi positif dan berekanada dibentangkan bagi mengekalkan ciri bentuk data masing-masing. Setiap skim diuji daripada banyak bentuk set data untuk menyokong kenyataan yang dibuat pembinaan berdasarkan skim tersebut. Selain itu, skim ini dibandingkan dengan bentuk konvensional splin nisbah kuadratik untuk menunjukkan kegunaan pembinaannya.

 

Kata kunci: Interpolasi; keselanjaran; pemeliharaan bentuk; splin nisbah trigonometri

REFERENCES

 

Akima, H. 1970. A new method of interpolation and smooth curve fitting based on local procedures. Journal of the ACM 17(4): 589-602.

Bashir, U. & Ali, J.M. 2013. Data visualization using rational trigonometric spline. Journal of Applied Mathematics 2013: Article ID. 531497.

Bashir, U., Abbas, M. & Ali, J.M. 2013. The G2 and C2 rational quadratic trigonometric Bézier curve with two shape parameters with applications. Applied Mathematics and Computation 219(20): 10183-10197.

Fritsch, F.N. & Carlson, R.E. 1980. Monotone piecewise cubic interpolation. SIAM Journal on Numerical Analysis 17(2): 238-246.

Gerald, C.F. & Wheatley, P.O. 2003. Applied Numerical Analysis. 7th ed. Boston: Addison-Wesley Publishing Company.

Goodman, T. 2001. Shape preserving interpolation by curves. Proceedings of Algorithms for Approximation IV. United Kingdom: University of Huddersfield. pp. 24-35.

Hussain, M.Z., Sarfraz, M. & Hussain, M. 2010. Scientific data visualization with shape preserving C1 rational cubic interpolation. European Journal of Pure and Applied Mathematics 3(2): 194-212.

Hussain, M.Z., Ayub, N. & Irshad, M. 2007. Visualization of 2D data by rational quadratic functions. Journal of Information and Computing Science 2(1): 17-26.

Hussain, M.Z. & Ali, J.M. 2006. Positivity-preserving piecewise rational cubic interpolation. MATEMATIKA 22(2): 147-153.

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

Kvasov, B.I. 2000. Methods of Shape Preserving Spline Approximation. Singapore: World Scientific Publishing Co.

Pan, Y.J. & Wang, G.J. 2007. Convexity preserving interpolation of trigonometric polynomial curves with a shape parameter. Journal of Zhejiang University Science A 8(8): 1199-1209.

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

Sarfraz, M. & Hussain, M.Z. 2006. Data visualization using rational spline interpolation. Journal of Computational and Applied Mathematics 189(1): 513-525.

Sarfraz, M. 2002. Visualization of positive and convex data by a rational cubic spline interpolation. Information Sciences 146(1): 239-254.

Schmidt, J. & Hess, W. 1987. Positive interpolation with rational quadratic splines. Computing 38(3): 261-267.

Schumaker, L.I. 1983. On shape preserving quadratic spline interpolation. SIAM Journal on Numerical Analysis 20(4): 854-864.

Zhu, Y., Han, X. & Han, J. 2012. Quartic trigonometric Bézier curves and shape preserving interpolation curves. Journal of Computational Information Systems 8(2): 905-914.

 

 

*Corresponding author; email: missheikh92@gmail.com