Construction of Cubic Timmer Triangular Patches and its Application in Scattered Data Interpolation

Abstract

This paper discusses scattered data interpolation by using cubic Timmer triangular patches. In order to achieve C1 continuity everywhere, we impose a rational corrected scheme that results from convex combination between three local schemes. The final interpolant has the form quintic numerator and quadratic denominator. We test the scheme by considering the established dataset as well as visualizing the rainfall data and digital elevation in Malaysia. We compare the performance between the proposed scheme and some well-known schemes. Numerical and graphical results are presented by using Mathematica and MATLAB. From all numerical results, the proposed scheme is better in terms of smaller root mean square error (RMSE) and higher coefficient of determination (R2). The higher R2 value indicates that the proposed scheme can reconstruct the surface with excellent fit that is in line with the standard set by Renka and Brown’s validation.

Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods

“…Furthermore, the proposed positivity-preserving scattered data interpolation is capable of producing a better interpolated surface than quartic Bézier triangular patches. This lies in contrast to scattered data schemes by Ali et al [1], Draman et al [9] and Karim et al [24], which are not positivity-preserving interpolations.…”

“…To apply the quartic triangular patch defined in Section 2.2 for scattered data, we use the local scheme comprising a convex combination between three local schemes K 1 , K 2 , and K 3 [1,9,24] such that:…”

“…Scattered data interpolation and approximation are still active research topics in computer-aided design (CAD) and geometric modeling [1][2][3][4][5][6][7][8][9]. This is because engineers and scientists often face the They applied the scheme for large scattered data sets to produce high-accuracy surface reconstruction.…”

Abel–Goncharov Type Multiquadric Quasi-Interpolation Operators with Higher Approximation Order

“…Furthermore, many researchers applied multiquadric quasiinterpolants to solve differential equations [15][16][17][18][19][20][21][22][23][24][25][26]. Meanwhile, Ali et al [27] constructed the SDI using Timmer triangular patches, which are used to visualize the energy data, i.e., spatial interpolation in visualizing rainfall data.…”

Geometric modeling and applications of generalized blended trigonometric Bézier curves with shape parameters

“…Yan expressed an algebraic-trigonometric mixed piecewise curve with two shape parameters and cubic trigonometric nonuniform B-spline curves with local shape parameters in [21] and [22], respectively. Hu et al [23] constructed geometric continuity constraints for H-Bézier curve of degree n. Recently, many researchers have developed the positivity-preserving rational quartic spline interpolation [24], cubic triangular patches scattered data interpolation [25], rational bi-cubic Ball image interpolation [26], quasiquintic trigonometric Bézier curve with shape parameters [27], curve modeling by new cubic trigonometric Bézier with shape parameters [28], continuity conditions for G 1 joint of S-λ curves and surfaces [29], generalized Bernstein basis functions for approximation of multi-degree reduction of Bézier curve [30], surface modeling in medical imaging by Ball basis functions [31], and geometric conditions for the generalized H-Bézier model [32] which have many applications in medicine, science, and engineering. Khalid and Lobiyal [33] presented the extension of Lupaş Bézier curves/surfaces and rational Lupaş Bernstein functions with shape parameters having all positive (p, q)-integers values.…”