Math in soccer

Date: 05/21/99 at 11:45:38
From: Elise Ricard
Subject: Math in soccer

I need to write a paper about how math is involved in soccer. So far, 
I have said some things like the field measurements and that they 
relate to area and perimeter. That is all I have so far.

Date: 05/24/99 at 13:41:09
From: Doctor Floor
Subject: Re: Math in soccer

Hi, Elise,

Thanks for your question!

You can read more about the soccer ball in the Dr. Math archives:

 http://mathforum.org/dr.math/problems/monaghan10.29.98.html   

The task of a goalkeeper also involves geometry:

Think of the following problem: if an attacking player is approaching 
the goalkeeper, where does the goalkeeper need to stand to have the 
best chance of preventing a score?

A good goalkeeper does not stand on the goal line too much of the 
time. He stays a bit in front of his goal, because it makes no sense 
to dive behind the goal line to "save" a ball. When a single attacker 
is approaching him, he will try to be on the angle bisector of the 
lines from that player to the goal posts. He will turn his body toward 
the approaching player, so that when he dives to the side to stop a 
shot, he is as far from the lefthand side as from the righthand side.


                    P---------goal---------P
                   /                   /
                  / ****GK         /
                 /        *    /
                /          /
               /       /
              /    /
             / /
          Attacker

The shortest distances from the goalkeeper (GK, on the not-drawn angle 
bisector) to the lines from the attacker to the goal posts are
perpendicular to these lines ("drawn" as ***). So the goalkeeper will 
not dive to the side, but to maximize his reach he will always dive 
slightly forward.

This is a simplification, because in the sketch above the attacker 
finds more space to curve the ball on the righthand side of the 
Goalkeeper (seen from the Attacker's viewpoint) than he finds on the 
lefthand side. Top players can use this to score very funny goals. I 
remember one by a Dutch player of Ajax Amsterdam (Peter van Vossen) a 
few years ago like this:


                                                P--------goal--------P 
                                          GK                     **** 
Van Vossen                                                  *****
           ********                                  *******
                   *************           **********
                                ***********

There also are interesting dilemmas for the goalkeeper about how far 
he should stand in front of his goal: when he is far from his goal the 
goalkeeper can get a single player's ball more easily, but the 
attacking player can also more easily lob the ball over the 
goalkeeper. And when a second player is coming at the same time, that 
player has a free path to the goal.

I hope you can use these ideas. Good luck on your paper!

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/