# Statistical Analysis of Table-Tennis Ball Trajectories

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{D}is an air drag coefficient, which can be measured, e.g., in wind-tunnel experiments.

_{L}as

^{−4}s) led to an end position even beyond the table. The difference of the final positions was 0.115mm for the Euler and RK4 methods. This was well below the precision of 1 mm we were looking for. However, the RK4 method was 4.37 times slower. Increasing the time step by a factor of five to compensate for the reduced calculation speed compared with the Euler method increased the error for the RK4 method beyond 1 mm, which we defined as our resolution limit.

^{5}random test trajectories with identical start conditions for both integrators. The differences are listed in Table 1.

^{−4}s.

^{8}initial conditions were sampled and trajectories calculated. Initially this was done on a Linux cluster with 32 cores. The run-time for each core was 20 h, resulting in a total run time of 640 h. Alternatively, GPU computing with CUDA was used on a Dell Precision T7500 desktop with NVIDIA Quadro FX3800. Here, only 3 h for the same calculation was needed. CUDA [8] is a programming interface using the parallel architecture of NVIDIA GPUs for general-purpose computing. CUDA library functions are provided as extensions of the C language, which allows for convenient and rather natural mapping of algorithms from C to CUDA. A compiler generates executable code for the CUDA device. The central processing unit (CPU) identifies a CUDA device as a multi-core coprocessor. For the programmer, CUDA consists of a collection of threads running in parallel. A collection of threads, which is called a block, runs on a multiprocessor at a given time. The blocks form a so-called grid. They divide the common resources, such as registers and shared memory, equally among them. All threads of the grid execute a single program called the kernel. All memory available on the device can be accessed using CUDA with no restrictions on its representation. However, the access times vary for different types of memory. Shared and register memory are the fastest, as they locate on the multiprocessor (on chip). The shared memory has the lifetime of the block and it is accessible by any thread on the block from which it was created. This enhancement in the memory model allows programmers to better exploit the parallel power of the GPU for general-purpose computing. Additionally, the texture memory that is off-chip allows for faster reading compared to global memory due to caching.

## 3. Results

_{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.

_{0}, ω

_{y}, and ω

_{z}. This combination led to a nearly equal distribution of eigenvalues, as shown in Table 5.

_{i,j,k,l}. Two dimensions from v

_{start}, ω

_{0}, ω

_{y}, and ω

_{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].

_{y}, but rather larger sidespin components ω

_{z}.

^{6})

^{2}× 4 B.

_{z}and ω

_{y}are visible in Figure 15 for the centers of the 40-mm case with increased net height and even stronger effects for the 44-mm ball, which had a smaller weight than the 40-mm ball. One needed larger spins to influence the trajectories of the 44-mm ball, because the effect of drag forces increased with larger size. The higher-net case needed stronger spins and reduced velocities for successful strokes. The ω

_{z}values were larger than the ω

_{y}contributions, meaning that stronger sidespin appeared, as discussed before.

## 4. Discussion

_{y}, but rather larger sidespin components ω

_{z}. This means that the game will not only slow down, but also diagonal play with longer reaction times for the opponent will get more important than fast parallel balls. One can also expect from this longer and more attractive rallies. However, the characteristics of the game will change strongly, because the possibilities for successful trajectories are limiting technical and tactical alternatives, reducing especially the influence of service.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Standard Deviation | Proportion of Variance | Cumulative Proportion | Eigenvalue | |
---|---|---|---|---|

PC1 | 2.705554 | 0.348573 | 0.348573 | 7.320027 |

PC2 | 1.633557 | 0.127072 | 0.475644 | 2.668508 |

PC3 | 1.508753 | 0.108397 | 0.584041 | 2.276336 |

PC4 | 1.400921 | 0.093456 | 0.677498 | 1.962581 |

PC5 | 1.108199 | 0.058481 | 0.735979 | 1.228106 |

PC6 | 1.008654 | 0.048447 | 0.784425 | 1.017382 |

PC7 | 1.001265 | 0.047740 | 0.832165 | 1.002531 |

PC8 | 0.998664 | 0.047492 | 0.879657 | 0.997331 |

PC9 | 0.994456 | 0.047093 | 0.926749 | 0.988943 |

PC10 | 0.906223 | 0.039107 | 0.965856 | 0.821241 |

PC11 | 0.519211 | 0.012837 | 0.978693 | 0.269580 |

PC12 | 0.474889 | 0.010739 | 0.989432 | 0.225520 |

PC13 | 0.288775 | 0.003971 | 0.993403 | 0.083391 |

PC14 | 0.235285 | 0.002636 | 0.996039 | 0.055359 |

PC15 | 0.188227 | 0.001687 | 0.997727 | 0.035429 |

PC16 | 0.154320 | 0.001134 | 0.998861 | 0.023815 |

PC17 | 0.124096 | 0.000733 | 0.999594 | 0.015400 |

PC18 | 0.079731 | 0.000303 | 0.999897 | 0.006357 |

PC19 | 0.034417 | 0.000056 | 0.999953 | 0.001185 |

PC20 | 0.028054 | 0.000037 | 0.999991 | 0.000787 |

PC21 | 0.014093 | 0.000009 | 1.000000 | 0.000199 |

Standard Deviation | Proportion of Variance | Cumulative Proportion | Eigenvalue | |
---|---|---|---|---|

PC1 | 2.656949 | 0.336161 | 0.336161 | 7.059382 |

PC2 | 1.625639 | 0.125843 | 0.462004 | 2.642702 |

PC3 | 1.501421 | 0.107346 | 0.569350 | 2.254266 |

PC4 | 1.402186 | 0.093625 | 0.662975 | 1.966126 |

PC5 | 1.158213 | 0.063879 | 0.726854 | 1.341458 |

PC6 | 1.028932 | 0.050414 | 0.777268 | 1.058701 |

PC7 | 1.002442 | 0.047852 | 0.825120 | 1.004891 |

PC8 | 0.998550 | 0.047481 | 0.872601 | 0.997102 |

PC9 | 0.990976 | 0.046763 | 0.919365 | 0.982033 |

PC10 | 0.900720 | 0.038633 | 0.957998 | 0.811297 |

PC11 | 0.555513 | 0.014695 | 0.972693 | 0.308595 |

PC12 | 0.439536 | 0.009200 | 0.981892 | 0.193192 |

PC13 | 0.435854 | 0.009046 | 0.990939 | 0.189969 |

PC14 | 0.250304 | 0.002983 | 0.993922 | 0.062652 |

PC15 | 0.224921 | 0.002409 | 0.996331 | 0.050589 |

PC16 | 0.177091 | 0.001493 | 0.997824 | 0.031361 |

PC17 | 0.171912 | 0.001407 | 0.999232 | 0.029554 |

PC18 | 0.105470 | 0.000530 | 0.999761 | 0.011124 |

PC19 | 0.054237 | 0.000140 | 0.999901 | 0.002942 |

PC20 | 0.040251 | 0.000077 | 0.999979 | 0.001620 |

PC21 | 0.021195 | 0.000021 | 1.000000 | 0.000449 |

Standard Deviation | Proportion of Variance | Cumulative Proportion | Eigenvalue | |
---|---|---|---|---|

PC1 | 2.673867 | 0.340455 | 0.340455 | 7.149567 |

PC2 | 1.641592 | 0.128325 | 0.468780 | 2.694827 |

PC3 | 1.534772 | 0.112168 | 0.580948 | 2.355527 |

PC4 | 1.401452 | 0.093527 | 0.674475 | 1.964069 |

PC5 | 1.122218 | 0.059970 | 0.734446 | 1.259375 |

PC6 | 1.016426 | 0.049196 | 0.783642 | 1.033122 |

PC7 | 1.002944 | 0.047900 | 0.831542 | 1.005897 |

PC8 | 0.999019 | 0.047526 | 0.879067 | 0.998039 |

PC9 | 0.996530 | 0.047289 | 0.926357 | 0.993072 |

PC10 | 0.909107 | 0.039356 | 0.965713 | 0.826476 |

PC11 | 0.524761 | 0.013113 | 0.978826 | 0.275374 |

PC12 | 0.426978 | 0.008681 | 0.987507 | 0.182311 |

PC13 | 0.339049 | 0.005474 | 0.992981 | 0.114954 |

PC14 | 0.241137 | 0.002769 | 0.995750 | 0.058147 |

PC15 | 0.195555 | 0.001821 | 0.997571 | 0.038242 |

PC16 | 0.155493 | 0.001151 | 0.998722 | 0.024178 |

PC17 | 0.130566 | 0.000812 | 0.999534 | 0.017048 |

PC18 | 0.084288 | 0.000338 | 0.999872 | 0.007105 |

PC19 | 0.037479 | 0.000067 | 0.999939 | 0.001405 |

PC20 | 0.031239 | 0.000046 | 0.999986 | 0.000976 |

PC21 | 0.017289 | 0.000014 | 1.000000 | 0.000299 |

Standard Deviation | Proportion of Variance | Cumulative Proportion | Eigenvalue | |
---|---|---|---|---|

PC1 | 2.637615 | 0.331286 | 0.331286 | 6.957013 |

PC2 | 1.676202 | 0.133793 | 0.465079 | 2.809653 |

PC3 | 1.571444 | 0.117592 | 0.582671 | 2.469438 |

PC4 | 1.400881 | 0.093451 | 0.676122 | 1.962469 |

PC5 | 1.130068 | 0.060812 | 0.736934 | 1.277054 |

PC6 | 1.028473 | 0.050369 | 0.787304 | 1.057757 |

PC7 | 1.007619 | 0.048347 | 0.835651 | 1.015296 |

PC8 | 0.999209 | 0.047544 | 0.883195 | 0.998420 |

PC9 | 0.998222 | 0.047450 | 0.930645 | 0.996447 |

PC10 | 0.867697 | 0.035852 | 0.966497 | 0.752898 |

PC11 | 0.521240 | 0.012938 | 0.979435 | 0.271691 |

PC12 | 0.394374 | 0.007406 | 0.986841 | 0.155531 |

PC13 | 0.333126 | 0.005284 | 0.992125 | 0.110973 |

PC14 | 0.246435 | 0.002892 | 0.995017 | 0.060730 |

PC15 | 0.221226 | 0.002331 | 0.997348 | 0.048941 |

PC16 | 0.163778 | 0.001277 | 0.998625 | 0.026823 |

PC17 | 0.131183 | 0.000819 | 0.999445 | 0.017209 |

PC18 | 0.087340 | 0.000363 | 0.999808 | 0.007628 |

PC19 | 0.046263 | 0.000102 | 0.999910 | 0.002140 |

PC20 | 0.037509 | 0.000067 | 0.999977 | 0.001407 |

PC21 | 0.022068 | 0.000023 | 1.000000 | 0.000487 |

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**Figure 1.**Air drag coefficient C

_{D}(upper green curve) and airlift coefficient C

_{L}(lower red curve) as a function of the ratio of spinning velocity u to the translational velocity v.

**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.

**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.

**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.

**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.

**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.

**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 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.

**Figure 13.**Cluster center coordinates of spin velocity versus start velocity for the four clusters for the different cases.

**Figure 14.**Cluster center coordinates of end velocity versus start velocity for the four clusters for the different cases.

**Figure 15.**Cluster center coordinates of side spin component ω

_{z}versus topspin component ω

_{y}of the normalized spin vector for the four clusters for the different cases.

**Table 1.**Absolute values of the differences between Euler and RK4 integrators for the 10

^{5}random test trajectories.

Averages | Maximum | Mean-Square Deviation | |
---|---|---|---|

x (m) | 6.360759 × 10^{−5} | 2.800000 × 10^{−4} | 5.157352 × 10^{−5} |

y (m) | 4.904501 × 10^{−4} | 1.120000 × 10^{−3} | 1.785499 × 10^{−4} |

z (m) | 4.464907 × 10^{−12} | 2.557150 × 10^{−11} | 3.791575 × 10^{−12} |

v (m/s) | 3.767089 × 10^{−4} | 2.600000 × 10^{−3} | 3.074972 × 10^{−4} |

v_{x} (m/s) | 7.743345 × 10^{−5} | 1.300000 × 10^{−3} | 8.600171 × 10^{−5} |

v_{y} (m/s) | 4.592264 × 10^{−4} | 2.800000 × 10^{−3} | 3.087756 × 10^{−4} |

v_{z} (m/s) | 7.391139 × 10^{−4} | 1.330000 × 10^{−3} | 2.169369 × 10^{−4} |

Case | Number of Successful Trajectories |
---|---|

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 |

Index | Variable | Comment |
---|---|---|

1 | x_{0} | x-position of starting point (varied between −3 and 0.3 m) |

2 | y_{0} | y-position of starting point (not varied; 0.381 m) |

3 | z_{0} | z-position of starting point (varied between −0.4m and 0.4 m) |

4 | v_{start} | absolute value of start velocity (between 20 km/h and 200 km/h) |

5 | v_{x;}_{0} | x-component of start velocity according to start angles |

6 | v_{y}_{,0} | y-component of start velocity according to start angles |

7 | v_{z}_{,0} | z-component of start velocity according to start angles |

8 | x_{e} | x-position of ball hitting the table |

9 | y_{e} | y-position of ball hitting the table |

10 | z_{e} | z-position of ball hitting the table (0 m) |

11 | v_{e} | absolute value of the final ball velocity |

12 | v_{x,e} | x-component of the final ball velocity |

13 | v_{y,e} | y-component of the final ball velocity |

14 | v_{z,e} | z-component of the final ball velocity |

15 | ω_{0} | absolute value of the spinning velocity (between 0 and 150 turns/s) |

16 | ω_{x} | x-component of the normalized spin vector |

17 | ω_{y} | y-component of the normalized spin vector |

18 | ω_{z} | z-component of the normalized spin vector |

19 | E_{kin,}_{0} | initial kinetic energy |

20 | E_{kin,e} | final kinetic energy |

21 | E_{rot} | rotational energy |

22 | z_{over net} | height of the ball over the net for a successful stroke |

23 | z_{max} | maximal height during a successful stroke |

**Table 4.**Principal component analysis (PCA) for the full set of variables: 40-mm ball reference case.

Standard Deviation | Proportion of Variance | Cumulative Proportion | Eigenvalue | |
---|---|---|---|---|

PC1 | 2.700487 | 0.347268 | 0.347268 | 7.292632 |

PC2 | 1.633091 | 0.126999 | 0.474267 | 2.666987 |

PC3 | 1.508682 | 0.108387 | 0.582654 | 2.276124 |

PC4 | 1.401304 | 0.093507 | 0.676161 | 1.963654 |

PC5 | 1.111800 | 0.058862 | 0.735023 | 1.236100 |

PC6 | 1.011844 | 0.048754 | 0.783777 | 1.023829 |

PC7 | 1.001355 | 0.047748 | 0.831525 | 1.002711 |

PC8 | 0.998771 | 0.047502 | 0.879027 | 0.997544 |

PC9 | 0.994223 | 0.047070 | 0.926098 | 0.988479 |

PC10 | 0.906493 | 0.039130 | 0.965228 | 0.821729 |

PC11 | 0.521662 | 0.012959 | 0.978186 | 0.272132 |

PC12 | 0.469405 | 0.010492 | 0.988679 | 0.220341 |

PC13 | 0.309721 | 0.004568 | 0.993247 | 0.095927 |

PC14 | 0.235542 | 0.002642 | 0.995889 | 0.055480 |

PC15 | 0.190701 | 0.001732 | 0.997620 | 0.036367 |

PC16 | 0.155887 | 0.001157 | 0.998778 | 0.024301 |

PC17 | 0.128646 | 0.000788 | 0.999566 | 0.016550 |

PC18 | 0.082267 | 0.000322 | 0.999888 | 0.006768 |

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 |

Standard Deviation | Proportion of Variance | Cumulative Proportion | Eigenvalue | |
---|---|---|---|---|

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 |

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## Share and Cite

**MDPI and ACS Style**

Schneider, R.; Lewerentz, L.; Lüskow, K.; Marschall, M.; Kemnitz, S. Statistical Analysis of Table-Tennis Ball Trajectories. *Appl. Sci.* **2018**, *8*, 2595.
https://doi.org/10.3390/app8122595

**AMA Style**

Schneider R, Lewerentz L, Lüskow K, Marschall M, Kemnitz S. Statistical Analysis of Table-Tennis Ball Trajectories. *Applied Sciences*. 2018; 8(12):2595.
https://doi.org/10.3390/app8122595

**Chicago/Turabian Style**

Schneider, Ralf, Lars Lewerentz, Karl Lüskow, Marc Marschall, and Stefan Kemnitz. 2018. "Statistical Analysis of Table-Tennis Ball Trajectories" *Applied Sciences* 8, no. 12: 2595.
https://doi.org/10.3390/app8122595