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### 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,

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/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Higher-Dimensional Geometry
Middle School Geometry
Middle School Higher-Dimensional Geometry
Middle School Two-Dimensional Geometry

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