The obvious mathematical concept involved in tennis would be counting. The scoring system in tennis is unusual as it doesn’t go up in units of one, or even units of the same amount. the follow picture depicts the different scoring options that could take place during one tennis game. As you can see, the first two scores (if won by the same person) go up in units of 15 (15-0, 30-0). However if the same person won the next point, one would presume it would be 45-0, however, due to the unusual scoring system, the new score would be 40-0.

Are there any deeper mathematical concepts embedded in the game of tennis?

Analysing statistics is a vital aspect for anybody involved in professional sport. Statistics can help players identify individual strengths and weaknesses and also the weaknesses of their opponents, which might inform their strategy and tactics for future matches. Percentages are the usual way of comparing players. In tennis, the typical statistics you see after each set include: first-serve percentage, service games won, break points saved, second-serve return points won, break points converted, unforced errors, forehand winners and so on.

 

 

http://www.changeovertennis.com/wp-content/uploads/2013/01/matchstatsmurraydjok.png

 

Before Andy Murray takes to the court to play a big semi-final against Djockovic, members of his training team will have analysed Djockovic’s match statistics from that specific tournament so see if they can identify any trends, likely predictions of play and any areas which are particularly weak so that Murray is stepping onto the court with informed information. This may not alter how Murray chooses to play, however if the match is not going as he would like, it would make sense that he would draw on the information he had been given to try and change up his tactics and play more aggressively to his opponents current weaknesses.

What about geometry?

Could tennis just be a problem of geometry? When playing a game of tennis, your aim is to win by the point by hitting the ball hard over the net, with spin, to a position where you opponent cannot return the ball. Is geometry involved in this process? In the path of the ball? The position of the player? The spin on the ball? It has been said by many commentators and some of the greats of the game that tennis is about controlling the middle of the court, cutting of angles and geometrical concepts support players in achieving this.

An attacking shot in tennis why relies on the play having a knowledge of angles is the angled approach shot. Hitting a short angled ball over the net forces your opponent forward, scooping up their return which should theoretically leave you with an easy volley to put away. You create a greater number of steeper angles when you step into the court and this forces your opponent to move around playing less confident and challenging returns. The following article discusses some of the geometric concepts in tennis further and provides some interesting food for thought for the next time you step out onto the tennis court.

Geometry and the Art of Tennis

What about the impact of top spin?

The ball bounce can greatly affect how an opponent returns your shot. Two properties which are involved in this process are Coefficient of restitution and Coefficient of kinetic friction. These two elements involve mathematical concepts such as ration, speed, distance and angles. The above diagram shows the bounce of a flat shot with no spin. The angle of incidence is approximately equal to the angle of reflection. This means the angle at which the balls impacts the ground and which the ball leaves the ground are almost the same. The ball slows down after it bounces due to the impact when hitting the surface.

The diagram above shows that when a ball is hit with top spin the angle of reflection is lower than the angle of incidence. The ball is not affected by the friction when impacting the surface of the court and the forward spin pushes the ball forward with greater speed after it bounces.

The following link is great for describing the effect different spins have on the ball bounce and how the different spins change the velocity, speed and direction of the ball.

The Physics of tennis