Monotone Data Modelling Using Rational Cubic Fractal Interpolation Function
Abstract
![](/public/site/graph.png)
Geometric modelling of several intricate and complex structures such as trees, mountains, clouds, ferns, geographic topography, and coastlines is challenging in computer graphics. Traditional splines such as
trigonometric, polynomial, exponential, and rational fail to simulate this significant class of complex structures, which are highly irregular in nature. For this purpose, this research develops a novel cutting-edge
method for synthesizing and modelling structures. The proposed technique; C 1 fractal interpolation function (FIF) builds an iterated function system (IFS) by integrating fractal calculus and rational cubic polynomial functions. Appropriate conditions on scaling and shape parameters are derived to help maintain the inherited shape qualities of the data. Experiments in numerous scientific domains, such as the pharmaceutical and chemical industries have been presented as an example, to confirm the usefulness of the suggested model. Moreover, the graphic results demonstrated that the developed monotone hybrid model (MHM) offers a heterogeneous method for gathering data with a monotone structure.
Downloads
References
Hussain MZ, Hussain M. Visualization of data preserving monotonicity. Appl Math Comput. 2007;190(2):1353–1364. https://doi.org/10.1016/j.amc.2007.02.022
Sarfraz M, Hussain MZ, Hussain M. Modeling rational spline for visualization of shaped data. J Numer Math. 2013;21(1):63–87. https://doi.org/10.1515/jnum-2013-0003
Hussain MZ, Sarfraz M, Hussain M. Scientific data visualization with shape preserving rational cubic interpolation. Eur J Pure Appl Math. 2010;2:194–212.
Chand AKB, Vijender N, Navescues MA. Shape preserving of scientific data through rational fractal splines. Calcolo. 2014;51: 329–362. https://doi.org/10.1007/s10092-013-0088-2
Cripps RJ, Hussain MZ. Monotone cubic Hermite interpolant. Appl Math Lett. 2012;25(8):1161–1165.
https://doi.org/10.1016/j.aml.2012.02.050
Floater MS, Pena JM. Monotonicity preserving on triangles. Math Comput. 2000;69(232):1505–1519.
https://doi.org/10.1090/S0025-5718-99-01176-X
Hussain MZ, Sarfraz M. Monotone piecewise rational cubic interpolation. Int J Comp Math. 2009;86(3):423–430.
https://doi.org/10.1080/00207160701595145
Barnsley MF. Fractal functions and interpolation. Constr Approx. 1986;2:303–329.
https://doi.org/10.1007/BF01893434
Barnsley MF, Harrington AN. The calculus of fractal interpolation functions. J Approx Theory. 1989;57:14–34.
https://doi.org/10.1016/0021-9045(89)90080-4
Balasubramani N. Shape preserving rational cubic fractal interpolation function. J Comput Appl Math. 2017;319:277–295. https://doi.org/10.1016/j.cam.2017.01.014
Arooj T, Hussain MZ, Hussain F. Scientific Data Visualization via Hybrid Model based on Fractal Spline Interpolation. Punjab Univ J Math. 2019;51(9):123–136.
Sarfraz M. A rational cubic spline for visualization of monotonic data. Comput Graph. 2000;24:509–516.
https://doi.org/10.1016/S0097-8493(00)00053-4
Sarfraz M. A rational cubic spline for visualization of monotonic data: an alternative approach. Comput Graph. 2003;27:107–121. https://doi.org/10.1016/S0097-8493(02)00249-2
Lamberti P, Manni C. Shape preserving functional interpolation via parametric cubics. Numer Algor. 2001;28:229–254.
Sarfraz M, Hussain MZ, Hussain M. Modeling rational spline for visualization of shaped data. J Numer Math. 2013;21(1):63–87. https://doi.org/10.1515/jnum-2013-0003
Beliakov G. Monotonicity preserving approximation of multivariate scattered data. BIT Numer Math. 2005;45(4):653–677. https://doi.org/10.1007/s10543-005-0028-x
Sarfraz M, Hussain MZ, Hussain F. Shape preserving curves using quadratic trigonometric splines. Appl Math Comput. 2015; 265:1126–1144. https://doi.org/10.1016/j.amc.2015.05.131
Hussain MZ, Hussain F, Sarfraz M. Shape preserving positive trigonometry spline curves. Iran J Sci Technol Trans Sci. 2018; 42:763–775. https://doi.org/10.1007/s40995-016-0056-1
Ahmad B, Perviaz A, Ahmad O, Dayan F. Numerical solution of fourth order homogeneous parabolic partial differential equations using non-polynomial cubic spline method (NPCSM). Sci Inquiry Rev. 2021;5(4):19–37.
https://doi.org/10.32350/sir/54.02
Ahmad B, Perviaz A, Ahmad MO, Dayan F. Solution of parabolic partial differential equations via non-polynomial cubic spline technique. Sci Inquiry Rev. 2021;5(3):60–76. https://doi.org/10.32350/sir/53.05
Copyright (c) 2023 Farheen Ibraheem
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.