Voronoi Diagrams in Football

Definition

A spatial partitioning of the pitch where every point is assigned to the nearest player, creating polygonal zones of influence. Each player "owns" the region of the pitch that is closer to them than to any other player, forming a tessellation that covers the entire surface.

Mathematical Origins

Voronoi diagrams are named after Georgy Voronoy (Ukrainian mathematician, 1868–1908), who formalized them in 1908. However, the concept has earlier roots: Peter Gustav Lejeune Dirichlet studied similar partitions in 1850, and the diagrams are sometimes called Dirichlet tessellations. In other fields they appear as Thiessen polygons (meteorology, after Alfred H. Thiessen, 1911) and in computational geometry as part of the Delaunay triangulation dual (Boris Delaunay, 1934).

The mathematical concept is straightforward: given a set of points (seeds) in a plane, the Voronoi cell for each seed is the set of all points closer to that seed than to any other. The boundaries between cells are equidistant from their respective seeds.

Application to Football

Voronoi diagrams were adapted for football analytics as tracking data became available in the 2010s. With 22 player positions known at each frame, computing a Voronoi tessellation shows which player controls (is closest to) each point on the pitch at any given moment.

The application gained visibility in the analytics community around 2018–2020, driven by:

• The Friends of Tracking YouTube series (featuring David Sumpter, William Spearman, Laurie Shaw, and others), which included tutorials on computing and visualizing Voronoi diagrams
• Open-source implementations shared by Laurie Shaw and other contributors
• Growing availability of tracking data samples (e.g., Metrica Sports open dataset)

Second Spectrum and other tracking data providers use Voronoi-based calculations internally as building blocks for more sophisticated spatial metrics.

How It Works

1. At each frame of tracking data, extract the x,y positions of all 22 players
2. Compute the Voronoi tessellation using standard computational geometry algorithms (Fortune's algorithm, O(n log n) complexity)
3. Clip the tessellation to the pitch boundaries
4. Color cells by team to show spatial control
5. Optionally compute cell area per player — larger cells indicate more space controlled

Modern implementations typically use scipy.spatial.Voronoi in Python or equivalent libraries.

What It Reveals

• Space between lines: gaps in a defensive block appear as large Voronoi cells for attackers in those areas
• Compactness: a team pressing high will have small, tightly packed cells in the opponent's half
• Isolation: a player surrounded by opponents will have a tiny cell, indicating they have no space
• Numerical superiority: in a given zone, the team with more (and smaller) cells has local dominance
• Transition moments: Voronoi diagrams in rapid succession reveal how space opens and closes during attacks

Limitations & Debates

Static Position Assumption

The fundamental limitation of basic Voronoi: it assigns space based solely on current position, ignoring player velocity, acceleration, and direction of movement. A player sprinting toward a zone at full speed effectively controls more of that zone than a stationary player who is technically closer. This is the primary motivation for the evolution toward Pitch Control Models.

Doesn't Account for Ball Position

Standard Voronoi treats all space equally, but in football, the space around the ball is far more contested and valuable than space on the opposite side of the pitch.

Computational Simplicity vs. Realism

Voronoi is computationally cheap and visually clear, which makes it appealing. But its simplicity means it's better as an intuitive visualization than as a precise analytical tool. Most professional tracking data companies have moved to more physics-based models internally.

Data Requirements

Requires tracking data (22 player positions per frame). Cannot be computed from event data alone, which limits public/amateur use. The Metrica Sports open dataset is one of the few freely available sources for experimenting with Voronoi.

Related Visualizations

• Pitch Control Models — the probabilistic evolution of Voronoi that accounts for velocity and acceleration
• Convex Hulls - Team Shape — complementary: convex hull shows team-level shape, Voronoi shows individual space control
• Pressing Maps — pressing compresses Voronoi cells in specific zones
• Heat Maps — aggregate spatial view vs. Voronoi's instantaneous snapshot

Key People

• Georgy Voronoy — mathematical formalization (1908)
• David Sumpter — popularized Voronoi in football context (Friends of Tracking)
• Laurie Shaw — open-source implementations and tutorials
• William Spearman — connected Voronoi thinking to Pitch Control Models|pitch control models

Notable Implementations & Resources

• scipy.spatial.Voronoi (Python) — standard library for computing tessellations
• mplsoccer (Python) — Voronoi plotting on pitch backgrounds
• Friends of Tracking YouTube series — tutorials and open-source code
• Metrica Sports open dataset — free tracking data for experimentation

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Tags: #football #analytics #visualization #voronoi #spatial-analysis