# Cubic \(q\)-Bezier Triangular Patch for Scattered Data Interpolation and Its Algorithm
## 5 Results and Discussion
### Testing on Seamount Real Data Points

In this subsection, we apply the proposed method to reconstruct real-life geologic data from the Seamount dataset. This dataset, obtained from MATLAB, represents the underwater surface of a seamount located on the Louisville Ridge in the South Pacific, mapped in 1984. Figure 31 illustrates the Delaunay triangulation of the Seamount data, comprising 294 data points, along with the corresponding 3D interpolation. In Figure 32, the surface interpolation of the Seamount data points is presented for various \(q\) values. Furthermore, Figure 33 showcases the 3D interpolation derived from Figure 32.

Table 5 provides a comparative analysis of CPU times for different \(q\) values based on the average of three experimental runs. Notably, the \(q\) value of 1.00 consistently results in the shortest average time of 157.99 s. This observation aligns with previous findings, reinforcing that a \(q\) value of 1.00 not only ensures computational efficiency but also maintains interpolation accuracy. Upon examining the visualizations, it is evident that the interpolations with a \(q\) value of 1.00 yield a smoother and more accurate representation of the Seamount surface compared to other \(q\) values. The reduced computational time further emphasizes the practicality of the proposed method for large and complex datasets. The following subsections will present a more detailed comparative analysis applied to engineering-related problems. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Function**} & \multirow{2}{*}{**Data Points**} & \multicolumn{2}{c}{\(\boldsymbol{R^{2}}\)} & \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** \\ \hline \multirow{3}{*}{\(F_{1}\)} & 36 & 0.9977 & 0.9976 & 0.0035 & 0.0036 & 0.1893 & 0.1986 \\  & 65 & 0.9992 & 0.9992 & 0.0020 & 0.0021 & 0.2166 & 0.2374 \\  & 100 & 0.9999 & 0.9999 & 0.0009 & 0.0008 & 0.2557 & 0.3051 \\ \hline \multirow{3}{*}{\(F_{2}\)} & 36 & 0.9829 & 0.9824 & 0.0130 & 0.0132 & 0.1758 & 0.1769 \\  & 65 & 0.9975 & 0.9973 & 0.0049 & 0.0052 & 0.1910 & 0.2638 \\  & 100 & 0.9988 & 0.9987 & 0.0035 & 0.0036 & 0.2807 & 0.3589 \\ \hline \multirow{3}{*}{\(F_{3}\)} & 36 & 0.9978 & 0.9977 & 0.0037 & 0.0038 & 0.1373 & 0.2090 \\  & 65 & 0.9995 & 0.9994 & 0.0018 & 0.0019 & 0.1570 & 0.2496 \\ \cline{1-1}  & 100 & 0.9999 & 0.9999 & 0.0007 & 0.0007 & 0.2078 & 0.4441 \\ \hline \end{tabular} \({}^{1}\) Proposed method; 2 [PERSON]’s method.

\end{table} 
Table 4: Overall error measurements for 36, 65, and 100 data points between the proposed method and [PERSON]’s method for \(q\)- Bézier with value of 1.00. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Function**} & \multirow{2}{*}{**Data Points**} & \multicolumn{2}{c}{\(\boldsymbol{R^{2}}\)} & \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** \\ \hline \multirow{3}{*}{\(F_{2}\)} & 36 & 0.9831 & 0.9824 & 0.0129 & 0.0132 & 0.2106 & 0.1640 \\  & 65 & 0.9973 & 0.9973 & 0.0051 & 0.0052 & 0.2550 & 0.1768 \\  & 100 & 0.9987 & 0.9987 & 0.0035 & 0.0036 & 0.2885 & 0.2968 \\ \hline \multirow{3}{*}{\(F_{3}\)} & 36 & 0.9977 & 0.9977 & 0.0038 & 0.0038 & 0.2145 & 0.1746 \\  & 65 & 0.9995 & 0.9994 & 0.0019 & 0.0019 & 0.2301 & 0.1817 \\  & 100 & 0.9999 & 0.9999 & 0.0008 & 0.0007 & 0.3059 & 0.2988 \\ \hline \end{tabular} \({}^{1}\) Proposed method; 2 [PERSON]’s method.

\end{table} 
Table 3: _Cont._Figure 31: Example for (**a**) true surface, (**b**) 3D interpolation, and (**c**) Delaunay triangulation of Seamount data points. Figure 32: Surface interpolation with different \(q\) values based on Seamount data points. (**a**) \(q=0.25\). (**b**) \(q=0.5\). (**c**) \(q=0.75\). (**d**) \(q=1\).