A physics-driven study of dominance space in soccer

In arXiv:2107.05714 the concept of the Voronoi diagram was investigated closely from a theoretical point of view. Then, a physics-driven kinematical method was introduced to produce an improved model for dominance space in soccer. Remaining faithful to the deterministic approach, we extend the original work 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. The asymmetric influence is fairly intuitive; players have more control in the direction they are running than 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. From simple kinematical laws, this effect can be quantified explicitly. For the frictional force, a portion comes from air resistance, and so will be proportional to the square of the player's speed, as is well known from fluid dynamics. There are no other external frictional forces, but, at the suggestion of biokinematics, there is an internal frictional force, relating to the consumption of energy by the muscles, which is proportional to the player's speed. Although these additions are intuitively understood, mathematically they introduce many analytical complexities. We establish exact analytical solutions of the dominance 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 dominance 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.