## 3 Results
To analyze the differences between the different cases a four-dimensional histogram of each case was calculated with the Numpy \"histogramd\" routine [24]. The relative deviation \(d\) was calculated with respect to the case of the 40-mm ball using the following equation:  
\[d_{i,j}^{case,40\text{mm}}=\sum_{k,l}\frac{\Phi^{case}-\Phi^{40\text{mm}}}{ \Phi^{40\text{mm}}+1}, \tag{4}\]  
where \(\Phi\) is the number of trajectories within the four-dimensional bin with the coordinates \((i,j,k,l)\) of the histogram; \(\Phi\) is an abbreviation for \(\Phi_{i,j,k,l}\). Two dimensions from \(v_{start}\), \(\omega_{0}\), \(\omega_{y}\), and \(\omega_{z}\) were selected for the plots denoted by \(i\) and \(j\), whereas the relative deviation was summed up over the remaining dimensions, \(k\) and \(l\). In order to account only statistically sensible data, differences in the numerator less than 10 were neglected. Plots were then generated with the Python packages Matplotlib [25] and HoloViews [26]. The 38-mm ball and the 40-mm ball were practically identical (Figure 5); very small differences showed up close to statistical variation limits. Figure 4: (left) Eigenvalues as a function of each principal component for the databases of the 38-mm ball and the 40-mm ball. (right) Values of the eigenvectors for the two cases. Figure 5: The two-dimensional (2D) distribution plots of the relative deviations as defined in the text of the databases for the 38-mm ball and for the 40-mm ball. The larger and heavier 44-mm ball showed clear deviations from the 40-mm reference database (Figures 6 and 7). The first eigenvector had an even stronger weighting of the spin compared to the 40-mm reference case. Also, the second and third eigenvectors increased the dominant weights for the translational velocity and the topspin, whereas the fourth eigenvector stayed unchanged. This change in eigenvectors can be easily understood. The global increase of successful strokes for the 44-mm ball compared to the 40-mm ball resulted mainly from the fact that the 40-mm ball was heavier (increasing the gravitational force). One needs larger spins to influence the trajectories of the 44-mm ball. For higher initial velocity, a larger number of successful shots was possible with the 44-mm ball, because the effect of drag forces increased with larger size. Deviations for the cases with 1- and 3-cm-higher nets for the 40-mm balls compared with the normal 40-mm scenario were more pronounced. With an increased net height of 1 cm (see Figures 8 and 9) and 3 cm (see Figures 10 and 11) for the 40-mm ball, a strong reduction in the number of successful strokes was detected. The reduction increased, as naively expected, when the net was higher. Figure 6: **(left)** Eigenvalues as a function of each principal component for the databases of the 44-mm ball and the 40-mm ball. **(right)** Values of the eigenvectors for the two cases. Figure 7: The 2D distribution plots of the relative deviations as defined in the text of the databases for the 44-mm ball and for the 40-mm ball. The first eigenvector had larger contributions from spin, but now also significant contributions from translational velocity. The second eigenvector had a reduced weighting of the translational velocity, but a larger weighting of the sidespin. The third eigenvector increased also the influence of sidespin, whereas the fourth eigenvector showed an increase of the spin contribution with a further reduction of the translational velocity. This was the result of the increased net height. If the starting velocity was too low, the ball was not able to pass the higher net successfully. Therefore, a reduction of the translational velocity was not possible. Figure 8: **(left)** Eigenvalues as a function of each principal component for the databases of the 40-mm ball with 1-cm-higher net and the 40-mm ball for standard net height. **(right)** Values of the eigenvectors for the two cases.