# Cubic \(q\)-Bezier Triangular Patch for Scattered Data Interpolation and Its Algorithm
## 5 Results and Discussion
This observation also highlights that different \(q\) values may yield the best results depending on the specific function and the nature of the real-life engineering problem being addressed. Another noteworthy observation is that different arrangements of points, even with the same number of points, yield similar Delaunay triangulation results, as illustrated in Tables 6 and 7. The performance metrics for these different point arrangements do not show significant variation, indicating that the spatial positioning of points, given a fixed number of points, is not a crucial factor in enhancing the performance of the proposed scheme. This observation suggests that the robustness of the Delaunay triangulation method is maintained irrespective of point arrangement. Figure 34: Example of electric potential of two point charges (Adopted from [38]). Figure 35: Delaunay triangulation of 25 data points (adopted from [39]). Figure 36: Surface and contour plot for (**a**) true function and (**b**) [PERSON] et al. [39]. Figure 37: Delaunay triangulation of 25, 35, 65, and 100 data points: (**a**) 25 data points; (**b**) 36 data points; (**c**) 65 data points; (**d**) 100 data points. Figure 38: Surface interpolation with 25, 36, 65, and 100 data points for \(q\) value of 1: (**a**) 25 data points; (**b**) 36 data points; (**c**) 65 data points; (**d**) 100 data points. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Data Points**} & \multicolumn{2}{c}{\(\boldsymbol{R^{2}}\)} & \multicolumn{2}{c}{**RMSE**} & \multicolumn{2}{c}{**CPU Time (s)**} \\ \cline{3-8}  & & \(\boldsymbol{Q^{1}}\) & \(\boldsymbol{Q^{2}}\) & \(\boldsymbol{Q^{1}}\) & \(\boldsymbol{Q^{2}}\) & \(\boldsymbol{Q^{1}}\) & \(\boldsymbol{Q^{2}}\) \\ \hline Fatin & 25 & 0.8958 & 0.8934 & 1.9374 & 1.9592 & 0.1244 & 0.0989 \\ \hline Proposed & 25 & 0.8868 & 0.8871 & 2.0195 & 2.0166 & 0.0743 & 0.1184 \\ \hline Proposed & 36 & 0.9356 & 0.9356 & 1.5233 & 1.5225 & 0.1463 & 0.1746 \\ \hline Proposed & 65 & 0.9791 & 0.9789 & 0.8674 & 0.8709 & 0.1954 & 0.2669 \\ \hline Proposed & 100 & 0.9908 & 0.9907 & 0.5753 & 0.5777 & 0.2696 & 0.3106 \\ \hline \hline \end{tabular}

* \(q=0.25\); \({}^{2}\)\(q=0.50\).

\end{table} 
Table 6: Overall error measurements for 25, 36, 65, and 100 data points for \(q\)-Bézier. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Data Points**} & \multicolumn{2}{c}{\(\boldsymbol{R^{2}}\)} & \multicolumn{2}{c}{**RMSE**} & \multicolumn{2}{c}{**CPU Time (s)**} \\ \cline{3-8}  & & \(\boldsymbol{Q^{3}}\) & \(\boldsymbol{Q^{4}}\) & \(\boldsymbol{Q^{3}}\) & \(\boldsymbol{Q^{4}}\) & \(\boldsymbol{Q^{3}}\) & \(\boldsymbol{Q^{4}}\) \\ \hline Fatin & 25 & 0.8901 & 0.8852 & 1.9898 & 2.0335 & 0.1654 & 0.1502 \\ \hline Proposed & 25 & 0.8881 & 0.8893 & 2.0079 & 1.9971 & 0.0715 & 0.0995 \\ \hline Proposed & 36 & 0.9361 & 0.9364 & 1.5173 & 1.5136 & 0.1160 & 0.1495 \\ \hline Proposed & 65 & 0.9786 & 0.9780 & 0.8776 & 0.8898 & 0.1889 & 0.1958 \\ \hline Proposed & 100 & 0.9905 & 0.9902 & 0.5836 & 0.5956 & 0.3125 & 0.2644 \\ \hline \hline \end{tabular}

* \(q=0.75\); \({}^{4}\)\(q=1.00\).

\end{table} 
Table 7: Overall error measurements for 25, 36, 65, and 100 data points for \(q\)-Bézier. Figure 39: Contour plot with 25, 36, 65, and 100 data points for \(q\) value of 1: (**a**) 25 data points; (**b**) 36 data points; (**c**) 65 data points; (**d**) 100 data points. Figure 42: CPU time (in seconds) for 36, 65, and 100 data points across three test functions, evaluated at four selected \(q\) values.