Convex Hulls — Team Shape & Compactness

Definition

In football analytics, a convex hull is the smallest convex polygon that encloses a set of player positions. When computed for a team's outfield players at a given moment, it represents the team's shape — the outer boundary of their spatial footprint on the pitch. The area of this polygon measures team compactness: a smaller area means the team is more compact, a larger area means they are more spread out.

Mathematical Origins

The convex hull is a fundamental concept in computational geometry, dating back to the 19th century. Informally, it's the shape you'd get if you stretched a rubber band around a set of points — the tightest enclosing polygon with no inward-facing angles.

Key algorithms for computing convex hulls include:

Graham scan (Ronald Graham, 1972) — O(n log n) complexity
Jarvis march (R.A. Jarvis, 1973) — also called the "gift wrapping" algorithm
Quickhull (Barber, Dobkin & Huhdanpaa, 1996) — the most commonly used in practice

In two dimensions (a football pitch), computing the convex hull of 10 outfield player positions is computationally trivial.

Application to Football

The application of convex hulls to football emerged as tracking data became available, primarily in the 2010s. Researchers and analysts used them to quantify tactical concepts that coaches had described qualitatively for decades: "stay compact," "don't stretch," "compress the space."

Javier Fernández and collaborators at FC Barcelona's Innovation Hub used convex hull area and related geometric measures in their spatial analysis work. David Sumpter discussed team shape metrics including convex hulls in Soccermatics (2016).

The concept gained wider visibility through the Friends of Tracking YouTube series and through tracking data companies like Second Spectrum and SkillCorner, who include compactness metrics in their analytical outputs.

Academic research on team compactness and shape has been published by several groups, including work by Daniel Memmert (German Sport University Cologne) and Matthias Kempe on spatial dynamics in team sports.

How It Works

Basic Computation

At each frame of tracking data, take the x,y positions of a team's outfield players (typically 10, excluding the goalkeeper)
Compute the convex hull — the smallest convex polygon enclosing all 10 positions
Calculate the area of this polygon (in square meters)
Optionally compute the perimeter, centroid (center of the team's shape), and aspect ratio (elongation)

Over Time

Track the convex hull area frame by frame to produce a compactness timeline over the match
Average area during defensive phases vs. attacking phases reveals tactical intent
Sudden changes in area correspond to transitions, pressing triggers, or defensive reorganization

Key Derived Metrics

Convex Hull Area

The total area enclosed by the team shape. Typical values:

Compact pressing team (defensive phase): ~400–800 m²
Spread attacking team (in possession): ~1200–2000 m²
These values vary significantly by team, tactic, and moment

Stretch Index

The maximum distance between any two players in the hull. Measures how "long" the team is. A high stretch index with low area means the team is stretched in one direction (e.g., a narrow defensive block).

Width and Length

Team width: horizontal span (touchline to touchline) of the hull
Team length: vertical span (goal line to goal line) of the hull
The ratio of length to width reveals team shape: narrow and deep vs. wide and flat

Centroid

The geometric center of the convex hull. Tracking the centroid's movement shows how the team shifts as a unit — moving up the pitch in possession, dropping back when defending, shifting left or right to cover overloads.

Effective Playing Space (EPS)

A refinement of convex hull area that weighs different zones differently (central areas weighted higher than flanks, areas near the ball weighted higher than far areas). Developed in academic contexts to better reflect the tactical relevance of the space a team occupies.

What It Reveals

Defensive Organization

A compact convex hull (small area) during defensive phases indicates a disciplined low or mid-block
Teams that press high will show a compact hull pushed high up the pitch (small area, high centroid)
A team losing shape (expanding hull) under pressure suggests disorganization

Attacking Shape

In possession, the hull typically expands as players spread to create space
Teams that play narrow (e.g., Manchester City under Guardiola) may have smaller attacking hulls than expected, with width provided by fullbacks only
The expansion rate after winning the ball indicates how quickly a team transitions into attack

Pressing and Transitions

Counterpressing: after losing the ball, a team that maintains a compact hull is positioned to counterpress. A rapidly expanding hull suggests the team is caught in transition
Defensive transitions: tracking hull area in the 5 seconds after losing possession reveals whether a team recovers shape quickly or is exposed

Comparing Teams

Convex hull area distributions over a season fingerprint a team's tactical identity
Atlético Madrid under Simeone: consistently compact defensive hulls
Guardiola's teams: controlled expansion in possession, rapid compression when pressing

Limitations & Debates

Goalkeeper exclusion: most implementations exclude the GK from the hull computation, which can miss situations where a sweeper keeper compresses the team's effective playing area
Convex hull oversimplifies: the convex hull is the outer boundary — it doesn't capture internal structure. A team could have the same hull area but very different internal distributions (clustered vs. evenly spaced)
2D limitation: doesn't account for the vertical structure of a team (lines of defense, midfield, attack). A flat 4-4-2 block and a stacked 3-1-4-2 could have similar hull areas but very different internal organization
Context dependency: hull area means different things depending on game state, score, and phase of play. A large hull while defending is bad; a large hull while attacking may be good
Requires tracking data: cannot be computed from event data, limiting public use
Alternatives gaining traction: some researchers prefer alpha shapes (a generalization of convex hulls that can be concave) or Delaunay triangulations for more nuanced team shape analysis

Related Metrics

PPDA - Passes Allowed Per Defensive Action — pressing intensity complements compactness (a compact team that also has low PPDA is pressing effectively)
Pitch Control Models — the most comprehensive spatial framework, which subsumes compactness information

Related Visualizations

Voronoi Diagrams — complementary: Voronoi shows individual space control, convex hull shows team-level shape
Pressing Maps — pressing effectiveness is related to team compactness
Heat Maps — aggregate version of spatial occupation, hull provides instantaneous shape

Key People

Ronald Graham — Graham scan algorithm for convex hull computation (1972)
Javier Fernández — spatial analysis at FC Barcelona using geometric team shape measures
David Sumpter — discussed team shape metrics in Soccermatics
Daniel Memmert — academic research on spatial dynamics in team sports
Matthias Kempe — research on team compactness and interpersonal distances

Notable Implementations & Resources

scipy.spatial.ConvexHull (Python) — standard library for convex hull computation
Friends of Tracking YouTube series — tutorials on team shape analysis
Memmert & Rein, research on spatial metrics in football
Second Spectrum / SkillCorner — commercial providers offering compactness metrics

Tags: #football #analytics #visualization #team-shape #compactness #spatial-analysis #convex-hull