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International Journal of Computer Science in Sport – Volume 12/2013/Edition 2 www.iacss.org

24

Computer simulations of table tennis ball

trajectories for studies of the influence of ball size

and net height

Ralf Schneider

1

, Oleksandr Kalentev

1

, Tatyana Ivanovska

2

& Stefan Kemnitz

3

1

Institute of Physics, Ernst-Moritz-Arndt University Greifswald

2

Institute of Community Medicine, Ernst-Moritz-Arndt University Greifswald

3

Faculty of Informatics and Electrical Engineering, University Rostock

Abstract

One possible measure to increase the medial appeal of table tennis is to slow down

the game by using bigger balls or higher nets. Usually, an empirical approach is

followed to study the effect of such changes on the players and the game. In this

work, a different approach is taken, namely solving numerically the equation of

motion for table tennis balls for systematical, statistical studies of the impact of

ball size and weight as well as of net height on the distribution functions of

successful strokes.

The analysis confirms the empirical observation that the change of the ball in the

year 2000 from a 38-mm to a 40-mm-ball can be compensated with other

parameters such that their resulting trajectory distribution functions are nearly

identical. This was also observed in reality, where adaptation of the player’s

technique compensated the larger ball size. A larger ball of 44 mm with small

weight is one option for suppressing high velocities, coupled also to a reduction of

the influence of spinning. As an alternative an increase of the net height is

possible. A small increase of the net height could be one future option, where the

basic character of the game is not strongly modified, but especially the influence

of the service could be reduced.

KEYWORDS: SPORTS EQUIPMENT, PHYSICS COMPUTING, MONTE CARLO

METHODS

Introduction

The medial appeal of table tennis seems to go down in terms of TV hours, at least outside Asia.

One of the reasons is the fact that the speed of the game is nowadays so high that it is very hard

for spectators to follow the balls (Nelson 1997, Djokic 2007). Possible counteractions to slow

down the game are to use bigger balls or higher nets. Usually, empirical studies are done to

study the effect of such changes on the players and the game. An alternative approach,

followed in this work, is the use of computer simulations. The equation of motion for table

tennis balls is solved numerically to allow systematical, statistical studies of the impact of ball

size and weight as well as of net height on the distribution functions of successful strokes.

One key problem for the medial appeal of table tennis is that the spin of the ball, the rotation, is

not visible for spectators, because they see only its effect. This makes it difficult to understand

why a simple looking ball of the opponent leads to a mistake for the other player. Therefore,

International Journal of Computer Science in Sport – Volume 12/2013/Edition 2 www.iacss.org

25

one intention of possible rule changes is to reduce the impact of spin on the game. Another goal

is to reduce the speed of the balls to allow a better visual tracking during the rallies (Djokic

2007). Some rule changes, like a larger ball, different counting system, stricter limits for

rubbers or new service rules, were already implemented and new modifications are under

discussion (Djokic 2007). For the players all rule or technical changes have strong impacts on

their techniques and strategies, requiring usually adaptations of their individual training

programs. Therefore, players are rather hesitant to new rules.

The 40-mm-ball played today is 2 mm larger and 0.2 grams heavier than the 38-mm-ball used

before. It has a larger air drag due to its larger cross sectional area reducing the maximum

velocities (Bai 2005). The mass distribution of the larger ball is shifted further away from the

center compared with the 38 mm ball. This creates a larger inertial moment and reduces the

spin. The larger 40-mm-ball results in a velocity and spin reduction of about 5 to 10 percent (Li

2005, Iimoto 2002). However, the larger ball had practically no impact on the characteristics of

table tennis, because larger exertions of forces by the players compensated the effects of the

size increase (Liu 2005, Li 2005). As a consequence of the modified technique, the fitness of

the individual player got more important. In modern table tennis the forces for a stroke are

created not only by the arms but the whole body is used to support this. A stronger athletics

allows more pronounced use of the legs producing larger forces on the ball, which are needed

to compensate the size increase. In addition, the wrist has to be used more effective to produce

spin. For the larger ball only the use of the forearm is no longer sufficient for spin, as it was the

case for the 38-mm-ball. The needs for larger exertion of forces amplify possible technical

mistakes, because the individual movement execution gets extended (Kondric 2007).

One obvious strategy to reduce the maximum velocity in table tennis rallies is to increase the

net height. However, such a change will have a severe impact on the characteristics of table

tennis, because this will limit very directly fast spins, shots and service. Therefore, up to now

this change of rule was avoided and ball size was the preferred correction action. Nevertheless,

a scientific data base is still missing for a decision.

In this work the impact of larger balls or higher nets on table tennis trajectories is studied using

computer simulations. A data base is created to quantify the influence of such changes.

Modifications in technique, tactics, strength and fitness are not considered in this analysis. For

a huge number of initial conditions the effect on successful strokes is studied. This delivers the

maximum amount of possible strokes for different conditions in terms of statistical

distributions which can be compared and analyzed. This represents already the best possible

adaptation to the changes, independent of what this would mean for the players in terms of

changes in their training. In particular the impact of the changes on the ball velocity

distributions will be discussed as motivated before.

After a short discussion of the effects of larger balls and higher nets as measures to slow down

table tennis, the forces acting on a moving ball are introduced. The computer code solving the

equation of motion is described and statistical analysis of trajectory distribution functions for

different balls and net heights is done. Using for this a GPU (Graphics Processing Unit) by

CUDA (Compute Unified Device Architecture, CUDA 2013) coding gives a very large speed-

up compared to CPUs. Results for different cases are compared and analyzed. Finally, the

results are summarized and discussed.

Methods

For a quantitative analysis of ball size and net height effects a computational approach is

followed. The basic element of the simulation is the solution of the equation of motion for table

International Journal of Computer Science in Sport – Volume 12/2013/Edition 2 www.iacss.org

26

tennis balls. The equation of motion needs a mathematical description of the acting forces. The

flight trajectory of a table tennis ball is determined by the gravitational force of the earth and

aero dynamical forces.

The gravitational force

gmF

G

⋅−=

r

alone results in a parabolic trajectory. This force acts towards the centre of the earth and

depends on the mass m of the ball and the gravitational constant g (9.81 m/s²).

The aero dynamical forces modify the simple parabola by air drag and lift. Air drag acts as

friction force against the direction of the movement of the ball. A simple example for this force

is the back pushing of a hand held out of a driving car. A larger velocity gives stronger force

acting against the direction of the car. This force also gets larger if one puts out not only a part

of the hand, but the full hand. It scales with the cross sectional area. The mathematical

expression is

vvACF

DD

⋅⋅⋅⋅⋅−=

ρ

2

1

r

,

with the density of air ρ, the cross sectional area A for a ball with radius r (

π

⋅

=

²rA ), the ball

velocity v and an air drag coefficient C

D

. This coefficient can be measured, e.g. in wind tunnel

experiments.

The second important aero dynamic force is the air lift. The so-called “Magnus effect”, named

after his discoverer Heinrich Gustav Magnus (1802-1870), is the reason that a rotating ball

experiences a deviation from its flight path. A famous example for this is a free kick goal from

the Brazilian soccer player Roberto Carlos in a friendly game with France at the 3

rd

of June

1997. Carlos gave a lot of spin to the ball during the free kick hitting the ball right from the

center of gravity with his left foot. The flight path of the ball got extreme passing around the

defenders who formed a wall into the goal.

The Magnus effect is a surface effect, because around the spinning ball a co-rotating air layer is

formed at the surface of the ball. The flying and spinning ball induces a pressure imbalance,

because on one side the ball is rotating with the air flow created by the movement of the ball in

the air, the other side opposite to it. On the side where counter-rotation exists, the total velocity

of the air flow is reduced, because both velocities compensate partly. On the co-rotation side a

larger flow velocity is created, because both velocities add up. Higher velocity in a flow means

lower pressure and the pressure differences on the two sides lead to the deviating Magnus

force, mathematically expressed with an air lift coefficient C

L

as

vevACF

LL

×⋅⋅⋅⋅⋅=

ω

ρ

2

1

r

The air lift force acts perpendicular to the axis of rotation

ω

e

and to the velocity

v

.

Air drag and lift coefficients of a rotating ball (see Figure 1) as a function of the ratio of

spinning velocity to translational velocity are implemented into the computer code as a fit of

experimental data (Achenbach 1972, Bearman 1976, Davies 1949, Maccoll 1928, Mehta 1985)

as a rational function y(x)

International Journal of Computer Science in Sport – Volume 12/2013/Edition 2 www.iacss.org

27

³²1

³²

)( xgxfxe

xdxcxba

xy

⋅+⋅+⋅+

⋅+⋅+⋅+

=

Figure 1: Air drag coefficient CD (upper green curve) and air lift coefficient CL (lower red curve) as a function

of the ratio of spinning velocity u to the translational velocity v.

During a topspin shot with forward rotation the lift force acts downwards, during a backspin

with backward rotation it acts upwards.

Swirling balls, often quoted in soccer and volleyball, can be created when the ball is hit with a

critical velocity leading to the access of the inverse Magnus effect. It shows up in Figure 1 for

low spinning velocities as a negative value of the air lift coefficient. This can lead also in table

tennis to swirling balls, because during the flight path the regime of positive and negative air

lift coefficients can change resulting in a swirling. However, for table tennis balls negative air

lift coefficients exist only where the coefficient itself is already quite small. Therefore, the

effect exists, but gives only deviations of some millimeters. The frequently quoted swirling

balls with long pimples are therefore more a psychological effect than physics: the pre-

programmed movement of the player anticipates a flight path of a strongly rotating ball from a

normal rubber sponge. The balls from the long pimples with reduced rotation have a different

flight path with less lift and fall down earlier such that the player is missing the ball and he

complains, that the ball was swirling.

The computer code solves the equation of motion of table tennis balls for given initial

positions, velocities and spins. An Euler solver was used, because its algorithmic simplicity

allowed an easy transfer onto the GPU with CUDA. A commonly used Runge-Kutta algorithm

was not chosen, because it has larger computational costs. A fourth order Runge Kutta

approach needs to calculate four times the forces, which slows down the code performance in

our case compared to the simple Euler method. This was not compensated by the larger time

step possible with the Runge-Kutta method compared to the Euler method. The dependence of

the aero dynamic forces on the velocity also does not allow the use of a Verlet algorithm.

Therefore, we decided to stay with the Euler method.

One example of a table tennis ball trajectory is shown as a red line in Figure 2. The table tennis

table region is marked in green, the net is blue. The orange sphere is the initial point of the

trajectory, where the ball is hit. The spinning of the ball is taken constant during the flight. x

International Journal of Computer Science in Sport – Volume 12/2013/Edition 2 www.iacss.org

28

and y are the spatial coordinates within the plane of the table tennis table. z is the height

coordinate above the table. A time step of 0.0001 seconds was used.

Figure 2: 3D trajectory of a table tennis ball

The ball in Figure 2 is hit at the baseline (x = 0 m) in the forehand part of the table (y = 0.6 m)

on the height of the table (z = 0 m). The black arrow shows the rotation axis of the ball, which

is here purely pointing into positive y-direction: the ball was a pure topspin without any

sidespin.

Results

For a statistical analysis of the effects of ball sizes and net heights on trajectories of table tennis

balls a Monte Carlo procedure was used. Many different initial conditions were solved: x was

varied between 0.3 m to -3 m, representing hitting locations from 30 cm above the table to 3m

behind the table. y was kept constant at 0.381 m, which is ¼ of the width of the table tennis

table. This was chosen as a representative position, the exact location of the hitting point in y

(forehand or backhand position) is not important for this numerical test. Initial height z was

sampled from 0.4 m to -0.4 m. The direction of the initial velocity was determined in the

following way: the horizontal angle was sampled between the limiting angles of the starting

point to the net posts, the elevation angle was chosen randomly. The spin axis was also

sampled randomly, that means topspin, backspin and sidespin were included.

The analysis was particularly aiming at fast shots. Therefore, only balls passing the net within

30 cm height distance were accepted. The absolute values of the translational velocities were

limited from 20 to 200 km/h, the spinning velocities from 0 to 150 turns/s (which is equal to

9000 turns/min). These values were determined empirically before as limits for 38 mm balls

(Wu 1993). These limits are probably different for other balls sizes and net heights, but in all

case studies successful hits were not restricted by the accessible parameter space chosen here.

A ball is counted as a successful ball if it passes the net within the height limit and hits the

other side of the table tennis table.

Monte Carlo studies using random numbers were done for the 38-mm-ball with a weight of 2.5

g, used in tournaments until end of 2000, the actual 40-mm-ball with 2.7 g and a 44-mm-ball

with a weight of 2.3 g, which was tested already in Japan. For the 40-mm-ball an increase of

the net height for 1 and 3 cm was analyzed, too.

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The sampling of such a large number of initial conditions guarantees to cover all possible

combinations of initial parameters (positions, translational and spinning velocities) for the

different cases creating a successful stroke. Clearly, for different balls and net heights the

parameter space of initial conditions leading to successful strokes will be different. The

database created in this study allows also an analysis of this effect.

For each case 5*10

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 hours resulting

in a total run time of 640 hours. Alternatively, GPU computing with CUDA was used on a Dell

Precision T7500 Desktop with NVIDIA Quadro FX3800. Here, only 3 hours for the same

calculation are needed. CUDA (CUDA 2013) is a programming interface to use 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 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, like 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’s 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 has been 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 which is off-chip

allows for faster reading compared to the global memory due to caching.

Our implementation consists of two main procedures. First, a predefined number of trajectories

are initialized on the CPU side. Thereafter, the ball movements are implemented on the GPU.

One step of the equation of motion for the ball’s trajectory, which includes the speed and the

position of the ball, is computed in a kernel. The input parameter of the kernel function is the

previous trajectory point. The calculations run for a maximal number of iterations. In each

iteration step, the updates of the ball’s position and velocity are computed, if the trajectory has

not stopped earlier, e. g., when the ball flew beyond the table.

Figure 3 shows as a function of initial position in y and z the number of successful trajectories.

The number of successful trajectories from half distance is nearly constant for all balls and net

heights. Only for distances below one meter the number of successful strokes decreases

continuously, because balls in this region have smaller probabilities hitting the table due to the

smaller angle. Balls hit above the table can again reach easier the other side. There is

practically no difference for the 38 and 40-mm-ball. Changes of the balls are compensated by

other parameter changes. The 44-mm-ball allows more successful strokes even for negative

height, because of its lighter weight and its higher air drag. A higher net affects strongly the

balls hits above the table limiting there the number of successful trajectories.

In general, the differences between the different cases get more pronounced the higher the

hitting point of the balls. A ball hit below the table must have a large spinning to reach the

other table side within the height limit. Larger velocities are not possible, because then the balls

are not able to reach the other table side and will pass beyond the baseline. Balls hit above the

table, even above the height of the net, can be hit with much higher velocities for a successful

strike.

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Figure 3: Number of successful trajectories as a function of initial y- and z-conditions

In Figures 4 and 5 the influence of the ball velocity on the distribution functions of the number

of successful strokes is shown. Figure 4 shows the dependence on the initial velocity, Figure 5

the dependence on the final velocity. The velocity range used for sampling the initial velocity

of 20-200 km/h is identical to 5.6-55.6 m/s.

Figure 4: Number of successful trajectories as a function of initial velocity of the balls

International Journal of Computer Science in Sport – Volume 12/2013/Edition 2 www.iacss.org

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Figure 5: Number of successful trajectories as a function of final velocity of the balls

Again, the results for the 38 and 40 mm ball differ only marginally. For the 44-mm-ball one

gets more successful trajectories compared to the 38 and 40-mm-ball for higher initial velocity,

the distributions for the final velocities are nevertheless very close again. However, very high

velocities above 35 m/s are suppressed earlier for the 44-mm-ball. A stronger influence is

visible for the 40-mm-ball increasing the net height. Already for smaller initial and end

velocities of about 10 m/s a reduction of successful trajectories shows up being equivalent to a

slowing-down of the game. For very low velocities the impact of the air drag is not yet

important resulting in larger number of successful trajectories.

Figure 6: Number of successful trajectories as a function of spinning velocity

Figure 6 demonstrates that the influence of spin is rather weak, because all differences are

within 20 percent. The number of successful trajectories is biggest for the 44-mm-ball,

followed by nearly identical numbers for the 38 and 40 mm ball and the case with a 1 cm

increase of the net height. As expected the highest net gives the smallest number of successful

International Journal of Computer Science in Sport – Volume 12/2013/Edition 2 www.iacss.org

34

trajectories. The ratio of successful trajectories with strong spinning to those with little

spinning is nearly the same in all cases with the exception of the 44-mm-ball. Here, the

influence of spinning on the distribution is strongly reduced.

Conclusions

Statistical analysis of the influence of ball size and net height on the number of successful table

tennis trajectories using computer modeling is used to quantify the effects on trajectory

distribution functions. The analysis confirm the empirical observation that the change of the

ball in the year 2000 from a 38-mm to a 40-mm-ball can be compensated such that their

resulting trajectory distribution functions are nearly identical. This was achieved in reality by

adaptations of the technique and the material. A larger ball of 44 mm with small weight is one

option for suppressing high velocities, resulting also in a reduction of the influence of spinning.

As an alternative option an increase of the net height is possible. For this, the character of the

game will change more strongly, because the possibilities for successful trajectories are

reduced limiting technical and tactical alternatives. A small increase of the net height could be

one option, where the basic character of the game is not too strongly modified, but reducing

especially the influence of the service.

Modifications of basic rules of table tennis like ball size and net height can reduce the

maximum velocities, but such modifications will be linked with severe changes in the

characteristics of table tennis: dynamics, technique and strategy will change strongly, too. The

question is if a possible gain in attractivity of table tennis for TV by such changes is worth the

loss of key elements of existing table tennis.

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