# Cubic \(q\)-Bezier Triangular Patch for Scattered Data Interpolation and Its Algorithm
## 5 Results and Discussion
\[F_{1}(x,y) =0.75e^{\left(-\frac{(9x-2)^{2}+(9y-2)^{2}}{4}\right)}+0.75e^{ \left(-\frac{(9x+1)^{2}}{49}-\frac{(9y+1)^{2}}{10}\right)}+\]7 Surface interpolation based on \(F_{3}\) function with 36, 65, and 100 data points when \(q=1\): (**a**) true surface; (**b**) 36 data points; (**c**) 65 data points; (**d**) 100 data points. \[F_{1}(x,y) =0.75e^{\left(-\frac{(9x-2)^{2}+(9y-2)^{2}}{4}\right)}+0.75e^{ \left(-\frac{(9x+1)^{2}}{49}-\frac{(9y+1)^{2}}{10}\right)}+\]8 Contour plots for 100 data points based on \(F_{2}\) function. (**a**) \(q=0.25\). (**b**) \(q=0.5\). (**c**) \(q=0.75\). (**d**) \(q=1\). \[F_{1}(x,y) =0.75e^{\left(-\frac{(9x-2)^{2}+(9y-2)^{2}}{4}\right)}+0.75e^{ \left(-\frac{(9x+1)^{2}}{49}-\frac{(9y+1)^{2}}{10}\right)}+\]9 Contour plots based on \(F_{3}\) function with 36, 65, and 100 data points when \(q=1\): (**a**) true surface; (**b**) 36 data points; (**c**) 65 data points; (**d**) 100 data points. \[0.5e^{\left(-\frac{(9x-7)^{2}+(9y-3)^{2}}{4}\right)}-0.2e^{\left( -(9x-4)^{2}-(9y-7)^{2}\right)}\]0 Surface interpolation for 36 data points based on \(F_{3}\) function. (**a**) \(q=0.25\). (**b**) \(q=0.5\). (**c**) \(q=0.75\). (**d**) \(q=1\). \[0.5e^{\left(-\frac{(9x-7)^{2}+(9y-3)^{2}}{4}\right)}-0.2e^{\left( -(9x-4)^{2}-(9y-7)^{2}\right)}\]1 Contour plots for 65 data points based on \(F_{3}\) function. (**a**) \(q=0.25\). (**b**) \(q=0.5\). (**c**) \(q=0.75\). (**d**) \(q=1\). \[0.5e^{\left(-\frac{(9x-7)^{2}+(9y-3)^{2}}{4}\right)}-0.2e^{\left( -(9x-4)^{2}-(9y-7)^{2}\right)}\]2 Surface interpolation for 100 data points based on \(F_{3}\) function. (**a**) \(q=0.25\). (**b**) \(q=0.5\). (**c**) \(q=0.75\). (**d**) \(q=1\). Since visualization alone is not sufficient to determine which \(q\) parameter yields the best interpolation results, error analysis was employed to provide a more robust comparison. Tables 1 and 2 present the overall measurements for 36, 65, and 100 data points with different \(q\) values for the proposed \(q\)-Bezier interpolation scheme. Based on these tables, it is evident that as the \(q\) parameter increases, there is an improvement in the interpolation quality, reflected by higher \(R^{2}\) values, a reduced RMSE, and faster CPU time. Specifically, the \(q\) values of 0.75 and 1.00 show significant enhancements, indicating a trend toward the properties of the classical Bezier curve when \(q\) parameter equals 1. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multirow{2}{*}{**Data Points**} & \multicolumn{2}{c}{\(R^{2}\)} & \multicolumn{2}{c}{**RMSE**} & \multicolumn{2}{c}{**CPU Time (s)**} \\ \cline{3-8}  & & **Q 1** & **Q 2** & **Q 1** & **Q 2** & **Q 1** & **Q 2** \\ \hline \multirow{3}{*}{\(F_{1}\)} & 36 & 0.9974 & 0.9974 & 0.0038 & 0.0037 & 0.0981 & 0.1734 \\  & 65 & 0.9990 & 0.9991 & 0.0023 & 0.0023 & 0.2621 & 0.2743 \\  & 100 & 0.9998 & 0.9998 & 0.0011 & 0.0010 & 0.4495 & 0.4800 \\ \hline \multirow{3}{*}{\(F_{2}\)} & 36 & 0.9832 & 0.9832 & 0.0129 & 0.0129 & 0.1554 & 0.1819 \\  & 65 & 0.9970 & 0.9971 & 0.0055 & 0.0053 & 0.2714 & 0.3210 \\  & 100 & 0.9986 & 0.9987 & 0.0037 & 0.0036 & 0.5213 & 0.4021 \\ \hline \multirow{3}{*}{\(F_{3}\)} & 36 & 0.9973 & 0.9975 & 0.0042 & 0.0040 & 0.1336 & 0.1562 \\  & 65 & 0.9993 & 0.9994 & 0.0021 & 0.0020 & 0.2817 & 0.2079 \\ \cline{1-1}  & 100 & 0.9998 & 0.9999 & 0.0011 & 0.0010 & 0.3653 & 0.3178 \\ \hline \hline \end{tabular} 1

\end{table} 
Table 1: Overall error measurements for 36, 65, and 100 data points for \(q\)-Bézier with values of 0.25 and 0.50. Figure 30: Contour plots for 100 data points based on \(F_{3}\) function. (**a**) \(q=0.25\). (**b**) \(q=0.5\). (**c**) \(q=0.75\). (**d**) \(q=1\). The error analysis further supports the visual observations: the results for 100 data points consistently outperform those for 36 and 65 data points. This improvement is expected, as a higher number of data points provides a more detailed representation of the underlying function, thus enhancing the interpolation accuracy.