The offside rule is the one of the most complicated in soccer and leads to some controversial refereeing decisions, can they be justified? In light of the recent English performance in the European Championships, we will also be looking at goal-line decisions.

As Euro 2012 has just passed and we approach the Olympic Football, we thought it would be a good time to review the decision making of football referees, and the physics and maths behind these decisions, and potential technologies that could help them in the future.

Firstly, let’s have a look at the offside rule. If we are to talk about the controversial decisions to do with the offside rule, we do first have to describe what the offside rule really is. The rule was designed in order to stop “goal-hanging” or a situation when a player just stands high up the pitch, waiting for an easy one-on-one with the goalkeeper.

FIFA, the world football governing body,  defines offside as when a player is in a offside position and:

It is not an offence in itself to be in an offside position.

A player is in an offside position if:

• he is nearer to his opponents’ goal line than both the ball and the

second-last opponent (usually the final outfield player).

A player is not in an offside position if:

• he is in his own half of the ﬁ eld of play or

• he is level with the second-last opponent or

• he is level with the last two opponents

A player in an offside position is only penalised if, at the moment the ball

touches or is played by one of his team, he is, in the opinion of the referee,

involved in active play by:

• interfering with play or

• interfering with an opponent or

• gaining an advantage by being in that position

In other words, the offside rule says that if a player on team A is closer to the goal he is shooting into than the ball and all but one of the team B players except one then the ball cannot be passed to him. Lets use a diagram:

At this point, player E on the blue team can pass to both player C and player B. This is because there are two of the opponent, red team players, A and D, between them and the goal.

Now player A has approached player E in order to tackle him, and player C has run forward. Player E can still pass to B, because the two red team players are still between him and the goal. However, now, player E cannot pass to player C. This is because player C is both in front of the ball, and there is only one red team player, in this case it could be the goalkeeper or an outfield player if, for some reason, the keeper is not back, between player C and the goal he is shooting towards. Now, there is one more possibility:

In this instance, Player E can in fact pass to both player B and player C. This is because, though player C has only one opponent player between him and the goal, he is behind the ball and therefore not in a offside position.

Offsides can be very game changing as they can cause a goal to be disallowed or cause an important play to be broken up. This means that a lot of the offside decisions are very controversial.

Despite the importance of offside decisions, we still rely on a reasonably unreliable system to catch them, linesmen. Two referees that run up and down the sidelines. However, this relies purely on human perception, and if there is one thing we are good at, it is being tricked by simple illusions. Because we only view the world from one point, what we really see are two 2-Dimensional images (or a 2 spacial, 1 temporal dimensional image, if you want to nit-pic), of the 3 (spacial) dimensional (and, 1 temporal dimensional) reality. This means that linesmen, if they are not completely in line with the last defender, can find it very difficult to call, or not call, a very close offside decision. A situation when a player may only be half a meter offside, there is one main problem that occurs with the linesmen’s perception. This is where maths comes into it:

When a linesman watches a match, the natural way for him to think of the pitch is a 2 dimensional, the only interactions and positioning that matters is from left to right. Therefore, the easiest way for a linesman to judge an offside is to see if a player who is shooting right is closer to the goal on the right than the closest defender:

 Linesman’s view
 Bird’s-eye view

The linesman can see in this situation that the blue player without the ball is obviously offside. He may be closer to the linesman but due to the reasonably sensible position the linesman has, there is no illusion.
However, lets look at a new situation, one where the blue player is still offside, but not by as much:
 Bird’s-eye view

Here you will see the linesman is not parallel to the last defender, but is instead at an angle. The black lines show where he is looking and the green lines show his sight line to the offside blue player, and the defending red player. Here emerges the problem. From the point of view of the linesman, the blue player is further to the left, therefore he sees this:

The blue player looks slightly onside, and the further the blue player is away from the linesman or the red player is towards the linesman, the more onside the player appears. This phenomena is reversed if the blue player is closer to the linesman’s sideline than the red player, and he will appear to be offside when he is in a perfectly legal position.

There is only one position the linesman can take for this situation to be completely avoided: if the linesman is perfectly in line with the last defender at all times. However, football can be a very fast paced game, and at times it can be very difficult for officials to keep up with world-class athletes. It is obviously also a lot harder to make decisions when running at pace, so this is not always feasible. Perhaps a better method would be a camera that moves up and down the byline, always in line with the last defender, sending images to a screen monitored by an official who then raises their flag if they see an offence.

One of the main reasons that offsides can seem so controversial is that those watching on TV have a better view than the linesmen a lot of the time. This makes it very obvious when they have made a mistake, and the mistakes that are made are usually made at the most critical times: when the last defender is running backwards whilst chasing after an attacker that is through on goal.

But after the attacker has successfully got the ball and is onside, he then has his next challenge: scoring a goal. This seems like it should be a fairly simple thing to officiate, but Geoff Hurst’s infamous goal in the 1966 World Cup final is still remembered, and there have been several other incidents where it has been unclear whether the ball has in fact crossed the line.

The first issue with determining whether a ball has gone in, particularly after it hitting the crossbar, involves the same parallax illusion that the linesman has to deal with when they are looking at offsides. It is very rare that the linesman will be standing level with the goal line when looking at such a situation, so they don’t have an angle from which they are able to determine whether the ball has gone in. To try to combat this, there have been experiments with officials that remain next to the penalty area so they are in a better position to make such decisions. However, as the incorrectly disallowed goal that Ukraine scored against England in their opening match of Euro 2012 shows, this does not always help.

In fact, in this video there was also an offside that was missed; the original pass from the Ukraine defender was offside (even though it is out of shot in this video), so perhaps these two poor decisions cancelled each other out.

The issue perhaps came from the fact that the official was quite close to the goalpost. This left him with the following view.

 Bird’s eye view
 Linesman’s view

The correct position for the linesman for this situation is actually just a little further back than the goalpost, to be precise the point where a ball would be at the moment the whole ball crosses the line and it counts as a goal. However, if the linesman was too close in this position, it could create the illusion that certain balls were across the line that weren’t.

We can reduce this to a geometric problem. There is an area just behind the goalpost that the linesman wants to see. However, in an ideal world, he wants the post to at least partially block his sight of any ball that has not completely crossed the line, so that he can determine whether he has or hasn’t. The following diagram shows the areas of the goal that the officials can see in different positions.

The further away the linesman is, the better he can see the critical point just over the line, and in this image, even the blue linesman can only see part of the ball, and it may appear to be behind the post. The green and red linesmen cannot see the ball at all, and definitely cannot award the goal!

From these positions, the linesmen can see almost every goal that does go in, but there is potential for them to incorrectly award a goal. The worst culprit for this is the green official, who could think that a ball has passed his near post when it is not even touching the line! Theoretically, the further back the linesman is, the more accurate a view of the goal he gets. To conclude, in order to make the most accurate decisions, the official should attempt to stand either in line with or just behind the post, as far back as his eyesight will allow him to. It then becomes a human compromise of at what distance he can no longer see effectively enough.

Before we finish off, let us first make our case for football officials to brush up on their physics!

Here is a disallowed goal from the 2010 World Cup, which Frank Lampard scored against Germany.

It is actually possible to tell from anywhere on the pitch that the ball was definitely over the line. The ball hits the bar, bounces, and then hits the bar again. When it hits the bar for the first time, it imparts backspin, a considerable amount of it considering the speed of the shot. This means that the ball will bounce backwards. For it then to hit the bar again after bouncing backwards, it must have been over the line at the point when it bounced. It seems that Frank Lampard ‘didn’t need to see a replay’ because his physics knowledge had already given him his answer!

Ned Summers and Theo Caplan

If you want to get in touch you can follow and mention us on twitter,@theaftermatter, email us at contactus@theaftermatter.com or search “The Aftermatter”on Facebook.

We hope you have enjoyed this post. If you have, then please check out our last two posts:

In light of the recent announcement by Cern, we have a post on what the Higgs Boson is and why we were looking for it, and what the future holds.

The Physics of Field Athletics – Hammer Throw, Angular Momentum,  why Hammer throwers spin before they throw, and what would happen if everyone in the world spun around at the same time.

If you have any requests for posts, please let us know! Contact details are above.