A physics-driven study of dominant regions in soccer

In [1], a physics-driven kinematical method was introduced to produce an improved model for dominant regions in soccer. Contrary to other similar attempts, the model maintains the deterministic nature of the Voronoi diagram forgoing any probabilistic notions.

 

Remaining faithful to the deterministic approach, we extend the work of [1] by the introduction of (a) an asymmetric influence of the players in their surrounding area, (b) the frictional forces to the players' motion, and (c) the simultaneous combination of both effects. Players have more control in the direction they are running than in any other direction. The sharper the turn they must make to reach a point on the pitch, the weaker their control of that point will be.

For the frictional force, a portion comes from air resistance [2], and so will be proportional to v2, where v is the velocity of the player, as is well known from fluid dynamics. There are no other external frictional forces, but, at the suggestion of biokinematics [3], there is an internal frictional force, relating to the consumption of energy by the muscles, which is proportional to v.

We establish exact analytical solutions of the dominant areas of the pitch by introducing a few reasonable simplifying assumptions. Given these solutions the new Voronoi diagrams are drawn for the publicly available data by Metrica Sports. In general, it is not necessary anymore for the dominant regions to be convex, they might contain holes and may be disconnected. The fastest player may dominate points far away from the rest of the players.

[1] C. J. Efthimiou, The Voronoi Diagram in Soccer: a theoretical study to measure dominance space, https://arxiv.org/abs/2107.05714.

 

[2] A. V. Hill, The Air-Resistance to a Runner, Proc. R. Soc., B102 (1927), pp. 380-385, https://doi.org/10.1098/rspb.1928.0012.

 

[3] K. Furusawa, V. Hill, and J. L. Parkinson, The dynamics of  “sprint” running,

Proc. R. Soc., B102 (1927), pp. 29-42, https://doi.org/10.1098/rspb.1927.0035.