# Cubic \(q\)-Bezier Triangular Patch for Scattered Data Interpolation and Its Algorithm
## 5 Results and Discussion
Additionally, the CPU time analysis reveals that higher \(q\) values lead to faster computations. The combination of a higher \(R^{2}\), lower RMSE, and reduced CPU time demonstrates the effectiveness of using a higher \(q\) parameter. The subsequent subsection will compare the proposed method with another state-of-the-art interpolation technique, focusing on the two best-performing \(q\) parameter values. ### Comparison with [PERSON] and [PERSON]'s Method

Based on the results from Tables 1 and 2, the optimal \(q\) values for the proposed interpolation method are between 0.75 and 1.00. To further validate the effectiveness of the proposed method, we compare it with the cross-derivative method of [PERSON] and [PERSON] (1966). Tables 3 and 4 present the overall error measurements between the proposed method and the cross derivative method for \(q\) values of 0.75 and 1.00 across three test functions. The result indicate that the proposed method outperforms the cross-derivative method, particularly when the \(q\) value is 1.00. Specifically, the proposed method achieves higher \(R^{2}\) values and lower RMSEs for all three test functions. For instance, with a \(q\) value of 1.00, the \(R^{2}\) values are consistently higher, and the RMSE values are lower, indicating a more accurate fit to the data. Although the differences in \(R^{2}\) and RMSE between the proposed method and the cross-derivative method are not significant, the CPU time required for the proposed method is significantly shorter. The proposed method only takes 0.2557 s, 0.2807 s, and 0.2078 s to construct the surface interpolation for the \(F_{1}(x,y)\), \(F_{2}(x,y)\), and \(F_{3}(x,y)\) functions, respectively. In contrast, the cross-derivative method requires substantially more time for similar tasks. These findings suggest that the proposed method not only provides a high degree of accuracy but also offers significant computational advantages. Given these promising results, the next subsection will explore the application of the proposed method on real-life data. \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 3** ** & **Q 4** ** & **Q 3** ** & **Q 4** ** & **Q 3** ** & **Q 4** ** \\ \hline \multirow{3}{*}{\(F_{1}\)} & 36 & 0.9976 & 0.9977 & 0.0036 & 0.0035 & 0.1966 & 0.1893 \\  & 65 & 0.9991 & 0.9992 & 0.0022 & 0.0020 & 0.2424 & 0.2545 \\  & 100 & 0.9999 & 0.9999 & 0.0009 & 0.0009 & 0.3431 & 0.2557 \\ \hline \multirow{3}{*}{\(F_{2}\)} & 36 & 0.9831 & 0.9829 & 0.0129 & 0.0130 & 0.2106 & 0.1758 \\  & 65 & 0.9973 & 0.9975 & 0.0051 & 0.0049 & 0.2550 & 0.1910 \\  & 100 & 0.9987 & 0.9988 & 0.0035 & 0.0035 & 0.2885 & 0.2807 \\ \hline \multirow{3}{*}{\(F_{3}\)} & 36 & 0.9977 & 0.9978 & 0.0038 & 0.0037 & 0.2145 & 0.1850 \\  & 65 & 0.9995 & 0.9995 & 0.0019 & 0.0018 & 0.2301 & 0.2583 \\ \cline{1-1}  & 100 & 0.9999 & 0.9999 & 0.0008 & 0.0007 & 0.3059 & 0.3024 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Overall error measurements for 36, 65, and 100 data points for \(q\)-Bézier with values of 0.75 and 1.00. \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}  & & **PS 1** ** & **GS 2** ** & **PS 1** ** & **GS 2** ** & **PS 1** ** & **GS 2** ** & **PS 1** ** & **GS 2** ** \\ \hline \multirow{3}{*}{\(F_{1}\)} & 36 & 0.9976 & 0.9976 & 0.0036 & 0.0036 & 0.1966 & 0.1670 \\  & 65 & 0.9991 & 0.9992 & 0.0022 & 0.0021 & 0.2424 & 0.2045 \\ \cline{1-1}  & 100 & 0.9999 & 0.9999 & 0.0009 & 0.0008 & 0.3431 & 0.3071 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Overall error measurements for 36, 65, and 100 data points between the proposed method and [PERSON]’s method for \(q\)-Bézier with value of 0.75.