The General Description of the Tennis Strokes. For math oriented players only!

[toly]

Anatoly Antipin

1. The General Description of the Tennis Strokes

The tennis is the game about movement. There is a lot of body activity, a racquet and ball motions etc. It is the common practice to use Vectors to describe any object motion.

Definition: The vector is a straight line segment whose length is magnitude and whose orientation in space is direction.

Mostly, I will examine a velocity of the different objects.

Definition: The velocity is the vector. Its magnitude defined as the speed and orientation in space is the direction of the moving object.

All these vectors stuff maybe is not really very important, but can be helpful for understanding different tennis techniques. I’ll try to keep the matter as simple as possible.

The main distinction among the different tennis strokes

Definition: Flat stroke – the ball is not rotating after the stroke.

Definition: Spin stroke (topspin, sidespin, backspin, etc) - the ball is rotating after the stroke.

Question: How can tennis player produce the strokes according to their definitions?

Answer: If and only if the velocity of the racquet VR (Figure. 1.1) is the perpendicular to the strings plane around impact spot, the stroke is flat. Or, in other words, if the racquet is moved in the direction of the perpendicular to the strings plane around impact spot, the shot will be flat.



Figure 1.1. Racquet velocity (VR) in the case of the flat shot. Coordinate axis OZ and vector VR are perpendicular to the strings plane

In all other cases, there would be spin strokes. Since, it is practically impossible to maintain the velocity of the racquet perpendicular to the strings plane, in reality, there are no pure flat strokes.

What happens if the velocity of the racquet VR (Figure. 1.2) has an arbitrary direction?

Accordingly to the linear algebra, every vector of the racquet velocity VR can be decomposed as a sum of two orthogonal (perpendicular to each other) components (vectors). The flat (or normal) component VF is the perpendicular to the strings plane and the spin (or tangential) component VS is the parallel to it.




Figure 1.2.The arbitrary vector of the racquet velocity (VR) along with its components (VF, VS). The strings plane exists in the XOY plane

The flat component is mostly responsible for the boll velocity after impact VB (its direction and speed).

The spin component provides the ball rotation, but with some strings interference can also change the velocity of the ball VB. It can happen due to the racquet strings could “bite” the ball or provide enough friction between the ball and strings and move the ball in direction of the spin component.

In case of the flat stroke, the spin component usually is equal to zero.

Something else about spin component

Since, the spin component VS is the vector, it can be also decomposed on two orthogonal components in the strings plane.

I assume that racquet stings bed is perpendicular to a ground. Then, the vertical spin component VSpinVer resides in plane perpendicular to the court’s ground. The horizontal spin component VSpinHor belongs to the plane parallel to the court’s ground (Figure 1.3).




Figure 1.3. Vector of the spin component along with its components

Relationship between defined vectors and common tennis slang

The upward vertical spin component VSpinVer brushes the ball up, makes topspin, and also can drag the ball upward.
The downward vertical spin component brushes the ball down, produces backspin, and pushes the ball down.
The horizontal spin component VSpinHoz makes right/ left sidespin and moves the ball to the right/left.

In general, the boll spins in the direction of the spin component VS (brushing ball direction) and moves in direction of the flat component VF (perpendicular to the strings plane).

Digging effect

Sometimes the spin component cannot produce any significant ball rotation but can change considerably the ball direction and speed.

For instance, if the point of contact is the sweet spot and the flat component VF has very high speed. The ball just digs into strings bed; the strings hug the ball very hard and do not allow creating any ball’s rotation during “digging” phase of the impact.
Then strings catapult the ball very fast and the spin component does not have enough time to rotate the ball.

However, while the ball is in digging and catapult phases the spin component still moves the racquet in its direction and as a result it varies the ball’s direction and speed.

[sureshs]

In Fig 1.2, VR cannot be decomposed into VF and VS perpendicular and parallel to the string plane unless VR lies in a plane perpendicular to the string plane.

[user92626]

WTH, I thought people played sports to get away from classroom. Nerds ruin everything!!!

[toly]

Originally Posted by sureshs

In Fig 1.2, VR cannot be decomposed into VF and VS perpendicular and parallel to the string plane unless VR lies in a plane perpendicular to the string plane.

Two thin orthogonal lines define the plane perpendicular to the string bed. The arbitrary vector VR lies in that plane. Think a little bit please!

[toly]

Originally Posted by user92626

WTH, I thought people played sports to get away from classroom. Nerds ruin everything!!!

Sometimes we need a strong jolt!

[sureshs]

Originally Posted by toly

Two thin orthogonal lines define the plane perpendicular to the string bed. The arbitrary vector VR lies in that plane. Think a little bit please!

That is what I am saying. If VR lies in that plane perpendicular to the strings, it is not an arbitrary vector, so it cannot be used to describe general situations.

[toly]

Originally Posted by sureshs

That is what I am saying. If VR lies in that plane perpendicular to the strings, it is not an arbitrary vector, so it cannot be used to describe general situations.

What about red vector?



There is nothing special regarding it. It is also arbitrary vector. Give me any vector’s coordinates and I construct orthogonal plane to the string bed and vector will lie in this plane. How do you think I draw these vectors? I don’t use any extraordinary methods.

[sureshs]

Yes got it now. The way you drew it, VF and VS cannot be as shown, if VR is going up, away and towards the tip of the racket.

[soyelmocano]

I've often imagined tennis and golf using images similar to this in my head. In my mind you have the initial velocity and launch angle. Of course the speed is mostly determined by how fast the racquet head is moving. Launch angle came be the same either with open orclosed face (and whatever angle in between). For example, to achieve a launch angle of 45 deg upwards, simply angle the racquet 45deg and swing straight along that path. However, one could get the same launch angle with a square face. Now a faster swing on a steeper plane is required to overcome the natural deflection angle.
Once the ball leaves the strings I imagine the denser cushion of air on the leading spin side pushing the ball in the opposite direction, and gravity pulling down.
I taught myself how to hit a draw in golf laying in bed at night. I had tried a lot of hitting, but it didn't work until I imagined and understood what was happening.
It really helps to understand the principles of physics to visualize shots should be. However, once you're on the court, just let it flow.

[FrisbeeFool]

Toly, I majored in math in college, and I disagree with pretty much every comment you've made about tennis strokes. Most of your comments seem to be about how players are wristing their grounstrokes, and vectors can prove it somehow. I think If anything is happening with the wrist at the end of the stroke, it's because of the preparation that happened earlier. In my mind, players have relaxed, fluid strokes, and their wrists are relaxed at the end. I'm not sure how vectors fit into all your wristy groundstroke arguments.

I question whether you understand even basics concepts surrounding vectors. Most of your posts on technique are so off the wall, then you will draw a vector and claim it somehow supports your post. I'm not seeing the connections.

[Uthree]

If you want to do a 3D analysis, probably more value in describing the players movement using the the 3 axis. Players tend to do to little and/or too much rotation around X,Y,Z axis. This would be a great way to understand the shot but doesn't seem to get much airplay around here.

[toly]

Originally Posted by FrisbeeFool

Toly, I majored in math in college, and I disagree with pretty much every comment you've made about tennis strokes. Most of your comments seem to be about how players are wristing their grounstrokes, and vectors can prove it somehow. I think If anything is happening with the wrist at the end of the stroke, it's because of the preparation that happened earlier. In my mind, players have relaxed, fluid strokes, and their wrists are relaxed at the end. I'm not sure how vectors fit into all your wristy groundstroke arguments.

I question whether you understand even basics concepts surrounding vectors. Most of your posts on technique are so off the wall, then you will draw a vector and claim it somehow supports your post. I'm not seeing the connections.

Thank you very much. I really appreciate any negative comments, but can you be more specific please? For example, what is wrong with vectors? IMO, they give us good visual representation of motions. Maybe you know something better?

[toly]

Originally Posted by Uthree

If you want to do a 3D analysis, probably more value in describing the players movement using the the 3 axis. Players tend to do to little and/or too much rotation around X,Y,Z axis. This would be a great way to understand the shot but doesn't seem to get much airplay around here.

I’m talking about 3D, see please post 1, fig.1.1 and fig1.2.
I simplified fig. 1.3 (2D), because I don’t want to upset people too much.

[toly]

Originally Posted by soyelmocano

I've often imagined tennis and golf using images similar to this in my head. In my mind you have the initial velocity and launch angle. Of course the speed is mostly determined by how fast the racquet head is moving. Launch angle came be the same either with open orclosed face (and whatever angle in between). For example, to achieve a launch angle of 45 deg upwards, simply angle the racquet 45deg and swing straight along that path. However, one could get the same launch angle with a square face. Now a faster swing on a steeper plane is required to overcome the natural deflection angle.
Once the ball leaves the strings I imagine the denser cushion of air on the leading spin side pushing the ball in the opposite direction, and gravity pulling down.
I taught myself how to hit a draw in golf laying in bed at night. I had tried a lot of hitting, but it didn't work until I imagined and understood what was happening.
It really helps to understand the principles of physics to visualize shots should be. However, once you're on the court, just let it flow.

Practically, there is no deflection angle in tennis for the reason that racquet strings work as catapult.

[sureshs]

End of thread? This is all there is to the general description of tennis strokes?

[toly]

Originally Posted by sureshs

End of thread? This is all there is to the general description of tennis strokes?

Yes, this is the end of thread, maybe because everything is extremely simple and tremendously clear.

[TheTsongaKid]

What are the relative percentages of Vf and Vs to maximize a heavy topspin ball with good pace as well?

[corners]

Digging effect

Sometimes the spin component cannot produce any significant ball rotation but can change considerably the ball direction and speed.

For instance, if the point of contact is the sweet spot and the flat component VF has very high speed. The ball just digs into strings bed; the strings hug the ball very hard and do not allow creating any ball’s rotation during “digging” phase of the impact.
Then strings catapult the ball very fast and the spin component does not have enough time to rotate the ball.

However, while the ball is in digging and catapult phases the spin component still moves the racquet in its direction and as a result it varies the ball’s direction and speed.

If the spin component of the swing is there, there will probably always be spin as a result. Please see this paper by R. Cross and take note of

2. Spin Generation

5. Racquet rotation

[Funbun]

Originally Posted by FrisbeeFool

Toly, I majored in math in college, and I disagree with pretty much every comment you've made about tennis strokes. Most of your comments seem to be about how players are wristing their grounstrokes, and vectors can prove it somehow. I think If anything is happening with the wrist at the end of the stroke, it's because of the preparation that happened earlier. In my mind, players have relaxed, fluid strokes, and their wrists are relaxed at the end. I'm not sure how vectors fit into all your wristy groundstroke arguments.

I question whether you understand even basics concepts surrounding vectors. Most of your posts on technique are so off the wall, then you will draw a vector and claim it somehow supports your post. I'm not seeing the connections.

I have to agree here. You made tennis look unnecessarily complicated here, toly.

For an observational standpoint, I think you'll have to incorporate much more than just vectors here. If a standard player were to look deep and study your post, they wouldn't get around to hitting very well without more information.

There's a lot more to tennis than just vectors. You have to bring physics into this. Why not discuss force? Or leverage? Momentum? Vectors only go so far into explaining something literally everybody on this forum does when hitting a tennis stroke.

The biggest problem I have with this post is that it focuses too much on the racquet itself. The player is the one who controls the racquet, and therefore hits the ball. How about you post something that discusses how to attain the maximum magnitudes for the vectors you described in your original post?

Taking a solely math-based approach to tennis is limiting. It's better if you expanded to physics instead; it'll be much more applicable. Tennis is a physics-based game, anyway.

[toly]

Originally Posted by TheTsongaKid

What are the relative percentages of Vf and Vs to maximize a heavy topspin ball with good pace as well?

Rod Cross in article, “Physics of the Tennis Kick Serve”, http://twu.tennis-warehouse.com/lear.../kickserve.php ,described procedure how to calculate ball’s spin.





The racquet velocity (VR) and its components (VF, VS).

“The amount of topspin is shown in Fig. 6 with the symbol S. Experiments and theoretical estimates both indicate that S is given to a good approximation by

S = 1.45VA

where S is the spin in rpm, V is the racquet head speed in mph and A is the approach angle in degrees.

For example, if A= 0 then S = 0 meaning that there is no spin generated at all. If V = 100 mph and A = 30 degrees then S = 4350 rpm. The amount of spin therefore increases with both the speed of the racquet head and the approach angle of the racquet head. The amount of spin also depends on the speed of the incoming ball.”