# 4-3-3 without the ball, 3-2-5 with it. Formations as transformations

Source: https://www.footballdatascience.co.uk/learn/formations-as-transformations
Published: 2026-09-28

> A formation is a set of player positions, and the change from the defensive shape to the attacking one is a transformation. Measure it and you can see how far the team moves, how much it stretches, and which player's job is different.

## The football question

The team sheet says 4-3-3. Watch the match and, when your team has the ball, it looks more like 3-2-5: a back three, two in midfield, five across the front. Lose the ball and it snaps back. **How do you measure a change of shape?**

Formations aren't fixed. They shift with possession, pressure, the score and the manager's instructions. Linear algebra gives a precise way to describe the shift.

## The concept

A formation is a **set of player positions**. On the same 105 m × 68 m grid as [the movement](/learn/player-movement-vectors) and [passing](/learn/passing-vectors) parts of this series, each player is a point (x, y), with x running up the pitch and y across it. Stack the ten outfield players' points and the whole shape is a 10 × 2 [matrix](/learn/team-as-a-matrix): one row per player, one column for x and one for y.

A change of shape moves every point:

$$(x,\ y) \;\longmapsto\; (x',\ y')$$

<div class="plain" markdown="1">
In plain football

- **(x, y)** is where a player stands in one shape, say without the ball.
- **(x′, y′)** is where he stands in the other, with the ball.
- **The arrow** is the **transformation**: the rule that takes the first shape to the second.
</div>

## Two shapes of one team

Here's a made-up team, attacking left to right. The hollow circles are its 4-3-3 without the ball; the filled ones are its 3-2-5 with it. The gold arrow is the left-back.

<figure class="rank-chart">
<div role="img" aria-label="A pitch showing ten players in a 4-3-3 without the ball, and dashed arrows to their positions in a 3-2-5 with the ball. The whole shape moves about 20 metres up the pitch and spreads wider. The left-back moves inside into central midfield.">

</div>
<figcaption>Made-up positions in metres. Without the ball: left-back (30, 8), centre-backs (25, 26) and (25, 42), right-back (30, 60), midfield (42, 20), (38, 34), (42, 48), front three (55, 10), (60, 34), (55, 58). With it: back three (40, 14), (38, 34), (40, 54); the left-back and the holding midfielder form a pivot at (52, 28) and (52, 40); the front five (78, 4), (72, 22), (82, 34), (72, 46), (78, 64).</figcaption>
</figure>

Three simple measurements describe each shape:

| | Without ball | With ball |
|---|---|---|
| Centre of the team | (40.2, 34.0) | (60.4, 34.0) |
| Length, back to front | 35 m | 44 m |
| Width, side to side | 52 m | 60 m |

<div class="plain" markdown="1">
In plain football

- **The centre** is the average of the ten players' positions. With the ball, the team pushes **20.2 m up the pitch**.
- **Length** is from the deepest player to the most advanced: the team stretches from 35 m to **44 m**.
- **Width** is from touchline side to touchline side: the team spreads from 52 m to **60 m**, to pull the opposition apart.
</div>

## Who moves furthest?

Each player's movement is a vector, new position minus old, exactly as in the movement part. The three biggest moves, and the smallest:

| Player | Move (m) | Distance |
|---|---|---|
| Left central mid | (30, 2) | 30.1 m |
| Right central mid | (30, −2) | 30.1 m |
| Left-back | (22, 20) | 29.7 m |
| Right-back | (10, −6) | 11.7 m |

The two central midfielders push furthest, 30 m up into the front five. The right-back moves least of all: he tucks in to become the right side of the back three. But the most interesting move is the left-back's. It's almost as long as the midfielders', and it goes in a completely different direction: **20 m infield**.

## The team-wide transformation

Most of that movement is the whole team doing the same thing: **push up and spread out**. We can write that as one transformation for everyone:

1. **Measure from the team's centre.** Take each player's position relative to the centre of the shape.
2. **Stretch.** Multiply the lengthwise part by 44 ÷ 35 ≈ 1.26 and the widthwise part by 60 ÷ 52 ≈ 1.15, the same stretch as the team's length and width.
3. **Move the centre.** Put the result around the new centre, (60.4, 34.0).

As matrix multiplication, the stretch in step 2 is:

$$\begin{aligned} &\begin{pmatrix} x' - 60.4 \\ y' - 34.0 \end{pmatrix} \\ &\quad = \begin{pmatrix} 1.26 & 0 \\ 0 & 1.15 \end{pmatrix} \begin{pmatrix} x - 40.2 \\ y - 34.0 \end{pmatrix} \end{aligned}$$

<div class="plain" markdown="1">
In plain football

- **The right-hand column** is where a player stood in the old shape, measured from the old centre of the team.
- **The 2 × 2 matrix** is the stretch: 1.26 times as long, 1.15 times as wide. The zeros mean going longer doesn't change width, and going wider doesn't change length.
- **The left-hand column** is where the rule says he should end up, measured from the new centre.
</div>

Apply that rule to every player and compare with where each one actually goes:

| Player | Rule says | Actually |
|---|---|---|
| Left winger | (79.0, 6.3) | (78, 4) |
| Striker | (85.3, 34.0) | (82, 34) |
| Left-back | (47.6, 4.0) | (52, 28) |

For the front three, the rule is within about 3 m. Push up and spread out is exactly their job. The rule is within about 13 m for everyone except one player. For the **left-back** it's **24 m** out. The rule sends him up the touchline, because that's what a stretched 4-3-3 does with a left-back. Instead he goes inside to join the holding midfielder.

That's the useful part. **The transformation describes the team's collective movement, and what's left over shows who has an individual job.** An inverted full-back is a tactical instruction, and the numbers pick it out without being told.

<details markdown="1">
<summary><span>Show the maths<small>Transformations, translations and the leftovers. Optional.</small></span></summary>

A **linear transformation** of the plane is multiplication by a 2 × 2 matrix. Some that describe team shape:

$$\text{stretch: } \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$$

$$\text{rotate by } \theta\text{: } \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$

A stretch with \(a > 1\) lengthens the team; \(b < 1\) makes it narrower and more compact. A rotation turns the whole shape, as when a back line pivots across to cover the ball on one side.

Moving the centre is a **translation**: adding a vector \(\mathbf{t}\), not multiplying by a matrix. Linear transformation plus translation is called an **affine transformation**:

$$\mathbf{x}' = M(\mathbf{x} - \mathbf{c}) + \mathbf{c}'$$

where \(\mathbf{c}\) and \(\mathbf{c}'\) are the centres of the two shapes. Here \(M\) is diagonal with \(a = 44/35\) and \(b = 60/52\).

The **residual** for each player is the gap between where he goes and where the rule sends him, \(\lVert \mathbf{x}'_{\text{actual}} - \mathbf{x}'_{\text{rule}} \rVert\). With these ten players the average residual is 9.4 m, against an average move of 22.1 m: the rule accounts for most of the movement, not all of it.

Flipping the pitch at half-time, so a team always attacks left to right, is also an affine transformation: \((x, y) \mapsto (105 - x,\ 68 - y)\), a half-turn about the centre spot.

We chose \(a\) and \(b\) from the length and width. A more careful method fits the matrix that makes the residuals as small as possible overall (least squares, or Procrustes analysis for shapes).

</details>

## Why it matters

Once shapes are matrices of positions, questions about them get numerical answers:

- **With and without the ball:** how much longer, wider and higher is the team in possession, on average?
- **Who changes role:** which players move furthest, and whose movement the team-wide rule can't explain?
- **Compactness:** how narrow does the team get without the ball, and does it stay that way late in games?
- **Opponents:** does the next opponent's full-back invert too? Compare his residual with our left-back's.

With tracking data you can do this for every second of a match: average the positions in each phase of play, then compare the shapes.

## Limitations

- **Made-up shapes.** Real average positions come from tracking or event data, and a whole match's average hides the moments that matter.
- **One matrix is a summary.** Real shapes also bend, with one side pushed up and the other held back. A single stretch can't show that; residuals, or a separate rule for each unit, can.
- **Max minus min is fragile.** One player pushed very high changes the length. The spread of positions (their standard deviation) is steadier for real data.
- **Players swap positions.** If the left winger and left-back switch, the maths needs to know who is who in each shape.
- **The opposition shapes you.** A team pinned back by a strong opponent isn't choosing its compact shape. Compare like with like.

## Try it yourself

Without the ball, a back four stands at (30, 8), (25, 26), (25, 42) and (30, 60). Without changing anything else, squeeze the team 20% narrower: multiply each player's distance from the middle of the pitch (y = 34) by 0.8. Where does each defender end up, and how wide is the back four now?

## Further reading

- [Linear transformations and matrices](https://www.3blue1brown.com/lessons/linear-transformations), 3Blue1Brown. A matrix as a rule that moves every point of the plane.
- [Affine transformation](https://en.wikipedia.org/wiki/Affine_transformation), Wikipedia. Linear transformations plus translations, with the matrix forms.
- [Procrustes analysis](https://en.wikipedia.org/wiki/Procrustes_analysis), Wikipedia. How to line up two shapes as closely as possible, and measure what's left over.