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Soccer Field Theory

Posted 3 years ago
3 Replies
6 Total Likes

Can there be computational soccer? This fall I was trying to explain to some kids that the most dangerous place for attackers to have the ball is right in front of the goal where they can get off a good shot. Then it occurred to me the second most dangerous place is where someone can pass it to the attacker, the third region would be where a pass can be made to the second region, etc.. (The goal is on the left.)

Partitioned field for kids

Here I took the dimensions of a under-8 field, being 30 yards by 40 yards with a 12 yard goal. With the assumption of a reliable pass of 5 yards, the field gets partitioned into many regions, possibly explaining some of the differences with adult soccer (longer passes can be considered reliable).

At first I tried to use Regions, e.g. RegionDistance, but I wasn't able to get that to work (Does anyone know how to do that?) and then tried this solution that splits the field into a bunch of points, makes a Graph object, then the Graph functionality splits the field up.

points = N[
  Flatten[Table[{i, j}, {i, 0, 40, 1/2}, {j, 0, 30, 1/2}], 1]];
edges = Flatten[
  Table[If[EuclideanDistance[points[[i]], points[[j]]] < 5, 
    If[points[[i, 1]] == points[[j, 1]], {i -> j, j -> i}, j -> i], 
    Nothing], {i, Length[points]}, {j, i + 1, Length[points]}]];
graph = Graph[Join[edges, Table[i -> "Goal", {i, 20, 42}]]];
Graphics[Map[{RandomColor[], Point[points[[#]]]} &, 
Complement @@@ Partition[Table[VertexList[NeighborhoodGraph[graph, "Goal", d]], {d, 10, 1, -1}], 2, 1]]]

Then I changed it so the "most dangerous" region has the requirement that the angle of the goal is at least 1/2 radians (if the goal's perspective is too narrow it is hard to get past the goalie).

Partitions taking into account the angle

These pictures are not too profound, but maybe someone can do a simpler or more aesthetic solution.

One can think of this as being a physics-inspired solution (field theory of soccer), where every possible pass on the field is taken into consideration. But maybe there are some classical mathematical solutions?

Speaking of physics, what would be the equipotentials be if the net were given a charge (e.g. the two nets could be capacitors) Normally the net is irrelevant but if it had a charge how could that affect a charged ball trying to get into the net)?

3 Replies
Posted 3 years ago

Hello Todd,

This is interesting. I think the danger zones vary according to skill. For example, at younger ages (say, under 12), it is pretty unusual for a team to have two successful passes in a row. At the pro level, I believe that 50% of goals come from set-plays.

As a soccer person, you might be interested in the history of world cup that I developed using Mathematica. The notebook is attached to this post (also can be found at the bottom of my page).

enter image description here


Dear @Mark Lawton, your World Cup app is great. We consolidated everything in a single comment, with the notebook attached (code was too large for a comment).

Thanks Mark for posting this summary of the World Cup.

I wonder about kids and passing, because I have seen them do it in practice. Maybe because the reliable distance of their passes is smaller, they need more players on the field. In the above dimensions, it is 7v7. Not enough players to put in every zone in the above diagram.

There are practical reasons why there are only seven players on kid's team, but it might be a percolation threshold type of thing where above a certain density of players they can pass the ball all around the field. Imagine 11 adult players on a field a kilometer long...the game breaks down, though different than a game with kid players.

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