Scattered data interpolation using quartic triangular patch for shape-preserving interpolation and comparison with mesh-free methods

Scattered data interpolation is important in sciences, engineering, andmedical-based problems. Quartic Bezier triangular patches with 15 control points (ordinates) can also be used for scattered data interpolation. However, this method has a weakness; that is, in order to achieve C1 continuity, the three inner points can only be determined using an optimization method. Thus, we cannot obtain the exact Bezier ordinates, and the quartic scheme is global and not local. Therefore, the quartic Bezier triangular has received less attention. In this work, we use Zhu and Han's quartic spline with ten control points (ordinates). Since there are only ten control points (as for cubic Bezier triangular cases), all control points can be determined exactly, and the optimization problem can be avoided. This will improve the presentation of the surface, and the process to construct the scattered surface is local. We also apply the proposed scheme for the purpose of positivity-preserving scattered data interpolation. The sufficient conditions for the positivity of the quartic triangular patches are derived on seven ordinates. We obtain nonlinear equations that can be solved using the regula-falsi method. To produce the interpolated surface for scattered data, we employ four stages of an algorithm: (a) triangulate the scattered data using Delaunay triangulation; (b) assign the first derivative at the respective data; (c) form a triangular surface via convex combination from three local schemes with C1 continuity along all adjacent triangles; and (d) construct the scattered data surface using the proposed quartic spline. Numerical results, including some comparisons with some existing mesh-free schemes, are presented in detail. Overall, the proposed quartic triangular spline scheme gives good results in terms of a higher coefficient of determination (R2) and smallermaximumerror (Max Error), requires about 12.5%of the CPU time of the quartic Bezier triangular, and is on par with Shepard triangular-based schemes. Therefore, the proposed scheme is significant for use in visualizing large and irregular scattered data sets. Finally, we tested the proposed positivity-preserving interpolation scheme to visualize coronavirus disease 2019 (COVID-19) cases inMalaysia. � 2020 by the authors.