# Cubic \(q\)-Bezier Triangular Patch for Scattered Data Interpolation and Its Algorithm
## 5 Results and Discussion
### Application to the Electric Potential of Two Point Charges

The scattered data interpolation method obtained in the previous subsection can be applied directly to the problem of the electric potential of two point charges proposed in [38]. The equation can be written as

\[V=\frac{1}{4\pi\kappa_{0}}\frac{g}{h} \tag{16}\]

where \(\kappa_{0}=8.8541878\times 10^{-12}\frac{C}{Nm^{2}}\) represents the permittivity constant. \(g\) is the magnitude of the charge in coulombs, whereas \(h\) is the distance from the particle in meters. Normally, the electric field of two or more particles is calculated using superposition. Based on Figure 34, the electric potential at a point due to two particles is given by

\[V=\frac{1}{4\pi\kappa_{0}}(\frac{g_{1}}{h_{1}}+\frac{g_{2}}{h_{2}}) \tag{17}\]

Figure 33: Three-dimensional interpolation with different \(q\) values based on Seamount data points. (**a**) \(q=0.25\). (**b**) \(q=0.5\). (**c**) \(q=0.75\). (**d**) \(q=1\). \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{\(q\) Values} & \multicolumn{4}{c}{**CPU Time (s)**} \\ \cline{2-5}  & **First Run** & **Second Run** & **Third Run** & **Average** \\ \hline
0.25 & 163.25 & 159.80 & 158.05 & 160.37 \\ \hline
0.50 & 160.45 & 158.11 & 157.78 & 158.78 \\ \hline
0.75 & 157.74 & 157.76 & 159.17 & 158.22 \\ \hline
1.00 & 157.88 & 158.37 & 157.73 & 157.99 \\ \hline \end{tabular}
\end{table} 
Table 5: CPU time comparison between different \(q\) values. where \(g_{1},g_{2},h_{1}\), and \(h_{2}\) are the charges of the particles and the distance from the point to the corresponding particle, respectively. In taking \(g_{1}=2\times 10^{-10}C\) and \(g_{2}=3\times 10^{-10}C\), which are positioned in the \(x\)\(y\) plane at points \((0.25,0,0)\) and \((-0.25,0,0)\), the scattered data interpolation that uses the \(q\)-Bezier triangular patch can solve the proposed problem. The existing Delaunay triangulation for the proposed problem is illustrated in Figure 35 as presented by [PERSON] et al. [39]. Figure 36 showcases the surface interpolation and its corresponding contour plot for both the true function and the existing scheme of [PERSON] et al. [39]. However, our observation indicates that their Delaunay triangulation could be enhanced for better interpolation accuracy by creating a more scattered data distribution. Another critical observation from Figure 36 is that the method of [PERSON] et al. does not approximate the problem accurately, as the results do not closely resemble the actual contour plot. To address these issues, we manually constructed our own Delaunay triangulation by carefully plotting scattered data with an uneven distribution, not only for 25 data points but also for 36, 65, and 100 data points, as shown in Figure 37. This extension aims to provide a more comprehensive comparative analysis of the proposed scheme. Figures 38 and 39 illustrate the surface interpolation and its contour plot when the \(q\) parameter is set to 1. Based on the figures, using 100 data points results in a smoother and more refined surface interpolation and contour plot that closely resembles the true function of the proposed problem. This pattern is consistent with previous sections of this study, where an increased number of data points provided more detailed information and features, better representing the proposed solution. Tables 6 and 7 present the error analysis comparing [PERSON] et al. 's method [39] and our proposed method across various \(q\) parameter values. Our proposed method achieves the best performance with 100 data points, showing an \(R^{2}\) value of 0.9908 and an RMSE of 0.5753 when the \(q\) parameter is 0.25. This is particularly interesting because, in contrast to the previous sections with different functions, our proposed method performed best when the \(q\) parameter was equal to 1. This discrepancy suggests the potential for further exploration of optimal \(q\) parameter values to enhance scattered data interpolation schemes.