## 3 Results
Figure 10: **(left)** Eigenvalues as a function of each principal component for the databases of the 40-mm ball with 3-cm-higher net and the 40 mm ball for standard net height. **(right)** Values of the eigenvectors for the two cases. Figure 9: The 2D distribution plots of the relative deviations as defined in the text of the databases for the 40-mm ball with 1-cm-higher net and for the 40-mm ball with standard net height. successful strokes appeared for low starting velocities compared to the standard net. The higher net needed stronger spins and reduced velocities for successful strokes. For very low velocities, the impact of the air drag was not yet important, resulting in a smaller reduction compared to higher velocities of about 30 m/s (or higher), where a strong reduction is observed. This reduction in the number of successful trajectories is equivalent to a slowing-down of the game. The strong reduction in the number of successful strokes was linked with rather small top spin components \(\omega_{y}\), but rather larger sidespin components \(\omega_{z}\). In addition, cluster analysis was applied to the data in order to dissect the properties of the most interesting cases. The datasets of the trajectories included roughly three million entries. A naive approach with an agglomerative clustering algorithm would easily exceed the memory of most computer systems. The agglomerative clustering with no constraints scaled quadratically in memory such that a crude estimate amounted to 36 TB = (3 \(\times\) 10\({}^{6}\))\({}^{2}\)\(\times\) 4 B. Therefore, balanced iterative reducing and clustering using hierarchies (BIRCH) clustering from the Python scikit-learn library [23] was used to reduce the number of data points to about 1100 points. This tool is rather memory-effective and designed for big datasets. The method generates a tree structure which can also be passed to other clustering algorithms. In our case, this was the agglomerative clustering from the same library [23]. The metric used was Ward linkage with Euclidean affinity. This linkage minimized the variance of the clusters being merged. From these reduced datasets, dendrograms were generated with the Python scipy library [27]. A dendrogram is a plot of a hierarchical binary tree. The height of each line represents the distance between different datapoints with the given metric. From the dendrograms (see Figure 12), one can visually deduce the number of clusters for each case. Cutting the tree at a given height with maximum variation of the metrics delivered the minimum number of clusters required for this case. For all cases, one can find four clusters as sufficient to represent the largest metric variations. The reduced datasets for all cases were further diminished with K-means clustering from scikit-learn [23] until four clusters remained. The centers of the clusters contained the condensed properties of the different systems and could be compared easily. The sizes of the symbols also represent the number of members of the clusters, showing that they were nearly identical. Figure 11: The 2D distribution plots of the relative deviations as defined in the text of the databases for the 40-mm ball with 3-cm-higher net and for the 40 mm ball with standard net height. Figures 13-15 show the most significant differences for the cluster centers of the four clusters for the different cases. The larger 44-mm ball showed higher initial velocities for all four cluster centers compared to the 40-mm ball, as shown also in previous work [8], because the 40-mm ball was heavier, as discussed before (Figure 13). \[d_{i,j}^{case,40\text{mm}}=\sum_{k,l}\frac{\Phi^{case}-\Phi^{40\text{mm}}}{ \Phi^{40\text{mm}}+1}, \tag{4}\]0 Cluster center coordinates of end velocity versus start velocity for the four clusters for the different cases. \[d_{i,j}^{case,40\text{mm}}=\sum_{k,l}\frac{\Phi^{case}-\Phi^{40\text{mm}}}{ \Phi^{40\text{mm}}+1}, \tag{4}\]1 Dendrogram of the 40-mm reference case.