## 3 Results  
Datasets of successful hits for the different cases (38-mm ball, 40-mm ball, 44-mm ball, and 40-mm ball with 1- and 3-cm-higher net) were created. Table 2 shows that, in terms of successful strokes, the 38-mm and 40-mm balls varied only marginally. The heavier 44-mm ball allowed for more strokes that were successful, whereas the higher net reduced this number. The list of dataset variables is presented in Table 3. \begin{table}
\begin{tabular}{c c} \hline
**Case** & **Number of Successful Trajectories** \\ \hline
38 mm & 2,795,262 \\
40 mm & 2,793,202 \\
44 mm & 3,282,767 \\
40 mm + 1-cm net & 2,672,572 \\
40 mm + 3-cm net & 2,470,891 \\ \hline \end{tabular}
\end{table} 
Table 2: Number of successful trajectories for the different cases. The data of the different cases were loaded into a Pandas [22] table. The data of the trajectories themselves were normalized, i.e., the mean was zero and the standard deviation was set to one. The two variables \(y_{0}\) and \(z_{e}\) had to be removed from the original data, since their standard deviation was practically zero. The initial \(y\)-position \(y_{0}\) was not varied and \(z_{e}\) was the height at the end of the trajectory, which was, by definition, zero for successful trajectories. For these datasets, the principal component analysis (PCA) from the Python package scikit-learn [23] was applied to the 21-dimensional parameter space. Table 4 shows the results for the 40-mm ball reference case; the other PCA results can be found in Appendix A. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Standard Deviation** & **Proportion of Variance** & **Cumulative Proportion** & **Eigenvalue** \\ \hline PC1 & 2.700487 & 0.347268 & 0.347268 & 7.292632 \\ PC2 & 1.633091 & 0.126999 & 0.474267 & 2.666987 \\ PC3 & 1.508682 & 0.108387  
A further reduction would result in a severe loss of variance in the data. Using more variables would lead to a strong drop of the next eigenvalue, demonstrating the reasonable limit of variables to this subset. This reduction in the number of variables to four was plausible because the start velocity, together with the rotation velocity and the rotation axis, determines most of the dataset variation. Only two components of the rotation axis were needed, because this was a unit vector and, from two components, the third component was pre-determined. The influence of starting positions and initial angles of the start velocity was much weaker and was reflected by the second drop in eigenvalues after the 10 th component. Using this subset of variables, a more detailed statistical analysis was then done. Comparing the results for eigenvalues and eigenvectors of the 38-mm ball and 40-mm ball databases gave nearly \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & **Standard Deviation** & **Proportion of Variance** & **Cumulative Proportion** & **Eigenvalue** \\ \hline PC19 & 0.035563 & 0.000060 & 0.999948 & 0.001265 \\ PC20 & 0.029352 & 0.000041 & 0.999989 & 0.000862 \\ PC21 & 0.015071 & 0.000011 & 1.000000 & 0.000227 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._  
Figure 3: Eigenvalues as a function of each principal component for the databases of the 40-mm ball and the 40-mm ball with 3-cm-higher net. \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & **Standard Deviation** & **Proportion of Variance** & **Cumulative Proportion** & **Eigenvalue** \\ \hline PC1 & 1.007513 & 0.253771 & 0.253771 & 1.015083 \\ PC2 & 1.001774 & 0.250888 & 0.504658 & 1.003551 \\ PC3 & 0.997023 & 0.248514 & 0.753172 & 0.994055 \\ PC4 & 0.993636 & 0.246828 & 1.000000 & 0.987312 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: PCA with a reduced set of variables for the 40-mm case. identical results (Figure 4). The first eigenvector was dominated by the contribution from the spin velocity, the second one by the translational velocity, the third one by the topspin component, and the fourth by the sidespin.