# Construction of new cubic Bézier-like triangular patches with application in scattered data interpolation

Authors
• 1 Universiti Teknologi PETRONAS, Seri Iskandar, Malaysia , Seri Iskandar (Malaysia)
• 2 Universiti Utara Malaysia, Sintok, Malaysia , Sintok (Malaysia)
• 3 University of West Bohemia, Plzen, Czech Republic , Plzen (Czechia)
• 4 Ton Duc Thang University, Ho Chi Minh City, Vietnam , Ho Chi Minh City (Vietnam)
• 5 Prince Sattam bin Abdulaziz University, Wadi Aldawaser, Saudi Arabia , Wadi Aldawaser (Saudi Arabia)
• 6 Cankaya University, Ankara, Turkey , Ankara (Turkey)
• 7 Institute of Space Sciences, Magurele-Bucharest, Romania , Magurele-Bucharest (Romania)
• 8 China Medical University, Taichung, Taiwan , Taichung (Taiwan)
Type
Published Article
Journal
Advances in Difference Equations
Publisher
Springer International Publishing
Publication Date
Apr 08, 2020
Volume
2020
Issue
1
Identifiers
DOI: 10.1186/s13662-020-02598-w
Source
Springer Nature
Keywords
This paper discusses the functional scattered data interpolation to interpolate the general scattered data. Compared with the previous works, we construct a new cubic Bézier-like triangular basis function controlled by three shape parameters. This is an advantage compared with the existing schemes since it gives more flexibility for the shape design in geometric modeling. By choosing some suitable value of the parameters, this new triangular basis is reduced to the cubic Ball and cubic Bézier triangular patches, respectively. In order to apply the proposed bases to general scattered data, firstly the data is triangulated using Delaunay triangulation. Then the sufficient condition for C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C^{1}$\end{document} continuity using cubic precision method on each adjacent triangle is implemented. Finally, the interpolation scheme is constructed based on a convex combination between three local schemes of the cubic Bézier-like triangular patches. The detail comparison in terms of maximum error and coefficient of determination r2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r^{2}$\end{document} with some existing meshfree methods i.e. radial basis function (RBF) such as linear, thin plate spline (TPS), Gaussian, and multiquadric are presented. From graphical results, the proposed scheme gives more visually pleasing interpolating surfaces compared with all RBF methods. Based on error analysis, for all four functions, the proposed scheme is better than RBFs except for data from the third function. Overall, the proposed scheme gives r2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r^{2}$\end{document} value between 0.99920443 and 0.99999994. This is very good for surface fitting for a large scattered data set.