A goal-aligned coordinate system for invasion games

3.1 Non-similarity transformation

Angles and distances are elementary spatial relations that need to be represented accurately to facilitate analyses involving running speed, passes, shots at goal, and many other uses of spatio-temporal data.

Any transformation of the plane that preserves angles and distance relations (up to scaling) is called a similarity transformation and obtained from a composition of translation, rotation, reflection, and scaling. If a physical pitch and the standardized dimensions of a data specification do not have the same aspect ratio, as is often the case, standardization necessarily involves a non-similarity transformation.

As a consequence, angles and distances determined on standardized spatial data will misrepresent the actual ones to a degree. This is the case, for instance, in StatsBomb’s event data where straightforward tests confirm that, for instance, the length and orientation attributes provided for a pass are determined from its start and end location after standardization. They are therefore distorted if the actual pitch does not have the 1.5 aspect ratio of the company’s 120 × 80 square yards standard.

Vendors using more obviously unreal pitch dimensions such as the 100 × 100 percentage distances used for Opta instead provide angle and distance information computed before standardization. While this can be more accurate, it constrains any subsequent analysis to exactly those spatial relationships the provider cared to include.

3.2 Non-homogeneous transformation

Even if the aspect ratio of a given pitch coincides with the standardized dimensions, i.e., scaling of the bounding rectangle is uniform, area variability conflicts with the fixed size of field markings.

Standardized expression of spatial predicates such as whether a location is inside the penalty area is supported by a fixed mapping of coordinates to markings such as goal posts, penalty marks, or the corners of the penalty areas. Typical examples are included in Fig. 3. Fixation of coordinates for certain markings may, in fact, sometimes be a byproduct of their use in playing field registration (Cuevas et al., 2020) rather than an attempt to guarantee interpretability.

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Fig. 3

Example distortions of a pitch sized 105 × 68m² arising from standardization including fixed points. Left: Fixing corners of penalty areas in StatsBomb’s system ensures that the penalty area is to scale, but other areas are enlarged. Right: Mapping corners of goal and penalty areas and penalty marks in Opta’s system leads to stretched peripheral intervals (scaling factor >1) of the horizontal coordinate and a compressed central interval (scaling factor <1). Due to additional fixed coordinates, the actual number of intervals is even larger, and corresponding non-linearities exist in the vertical dimension. Both issues arise in all such systems and are indeed unavoidable when pitch sizes can vary while some markings do not.

No matter the motivation, fixed locations of field markings imply that different areas of a pitch that is not to scale of a data provider’s specifications will be transformed non-homogeneously.

To rule out the possibility of uniform scaling of the entire pitch, it suffices to consider a penalty area. As an example of a space with fixed dimensions, consider the 16.5m × 40.32m= 665.28m² penalty areas. They make up a variable percentage of the area of pitches with different size. At Coliseum Alfonso Pérez, the home ground of Getafe CF, they each covered 9.05% of the pitch which was 105 × 70m² at the time of the 2010 Women’s Champions League Final (cf. Table 1). The 18 × 44 penalty areas of StatsBomb’s 120 × 80 standardized pitch each cover only 8.25%, despite the common 1.5 aspect ratio of the actual and the standardized ground.

To see that the issue arises not just with areas but even within a single dimension, consider fixed coordinates for penalty marks. In Stats Perform’s Opta specifications, for instance, penalty marks are fixed at coordinates 〈11.5, 50〉 and 〈88.5, 50〉. A difference of 11.5 units in the first coordinate therefore represents a length of 11m. Applying the scaling factor of 11 11.5 to the 100 unit pitch length yields a pitch that is 95.65m long. Not only is this an uncommon pitch length, it is also contradictory to the locations assigned to corners of the penalty areas: here, a difference of 17 units represents a length of 16.5m, so that uniform scaling results in a factor of 16.5 17 > 11 11.5 . Stats Perform therefore resort to piecewise linear transformations in each dimension (Stats Perform, 2023, Appendix 7), which clearly affect angles.

While piecewise linear and more general non-homogeneous schemes reduce the representation error of point locations, no transformation of the plane can preserve angles, areas, and distances across different pitch sizes when coordinates of field markings are fixed, which in turn is desirable for interpretable coordinates (Criterion C2) and simple spatial predicates (Criterion C3).

Common specifications for the temporal dimension avoid the above issues because the only transformation they apply is translation by an offset.

Time is generally specified in multiples or fractions of a second, i.e., using the base unit of time in the International System of Units (SI), and with reference to either Universal Coordinated Time (UTC) or relative to the start of a match period. Given the starting times of match periods, i.e., an alignment of match time with actual time, temporal scales and reference points are easily transformed into each other without distortion or loss of information. Note that these starting times take the role of origins in a one-dimensional coordinate system, and that there is one such origin per match period.

If the temporal dimension was specified in the same way as the spatial, the duration of each period would be rescaled separately to fit a fixed time interval. By the 7th Law of the Game, a standard match lasts for two equal halves of 45 minutes, but at the discretion of the referee an allowance is made for time lost during each period. Similar to the variance in pitch dimensions, this results in a variable amount of added time, until recently about 2–7 minutes. If time were treated the way space is, the actual duration of each period would be rescaled (compressed) to fit an interval of the corresponding standard length.

The key decisions to note are that only rigid transformations (i.e., time shifts) are applied and that distinct reference points are used for each match period (i.e., their starting times).

4.1 Conventions

Invasion games are territorial and revolve around two special locations, typically the centers of two goals or other marks defended and attacked by either team. Therefore, the field of play generally has two symmetry axes, a longitudinal one intersecting the goals, and a lateral one perpendicular to it. The (opposing) directions of play for the two teams align with the longitudinal axis, because the teams are trying to close in on the other team’s goal. As a consequence, all common field-oriented spatial notions such as left and right, forward and backward, or high and low, align with the longitudinal axis when oriented according to the perspective of a team.

In football, it seems that the more frequent spectator perspective from the touchlines, which is also the (supposedly) non-partisan perspective of television broadcasts, may have led data providers to prefer horizontal pitch orientations and standardize only the direction of play, often from left to right. While the wider aspect ratio may also be a form factor on monitors and in print, vertical pitch orientation is more common in coaching materials (Wade, 1967; Zauli, 2003; Teoldo et al., 2022) and team meetings, because it reduces the dissonance between orientation during briefings and in matches.

In line with spatial terminology, tactical practice, and physical measurement we therefore adopt the following conventions:

Vertical pitch orientation with the two goals at the bottom and top.

4.2 Definition

As argued above, we assume, without loss of generality, vertical orientation and a team-specific perspective. We next define a coordinate system that represents a location on the pitch not in one but two, complementary, ways. This redundancy is conceptual only, and does not affect storage requirements.

Since there is no single point of reference from which distances are invariant on pitches of different size, we instead use two reference points that together make every pitch location addressable in an interpretable way, independent of pitch size.

In invasion games, no two points are more characteristic references than the centers of the goals. With these as the origins of two related (local) Cartesian coordinate systems, we obtain two different expressions for the same location: while the x-coordinate referring to the lateral dimension is shared, the y-coordinate is either with reference to a team’s own goal line or the opposition’s goal line. In the following definition, the two reference points should be thought of as the goal defended and the goal attacked.

Definition 1. Let G ¯ , G ¯ be two reference points in a two-dimensional Cartesian coordinate system. A goal-aligned coordinate system is obtained from twin transformations each composed of

a rotation that aligns the vertical axis with the vector from G ¯ to G ¯ and

A single location is thus equivalently expressed as either 〈 x , y _ 〉 in the coordinate system centered at goal G ¯ (e.g., during moments when the team is defending it) or 〈 x , y ¯ 〉 (e.g., when the team is attacking the opposition’s goal G ¯ ). The two expressions are related by y ¯ ℓ y ¯ = ℓ , where ℓ is the distance between the two goal lines, i.e., the length of the pitch. As a consequence, y ¯ < y ¯ and y ¯ ≡ y ¯ mod ℓ always. Given the contextual parameter ℓ, each of the two expressions can be reconstructed from the other, so only one needs to be stored. In the following, we may hence assume that all three coordinates are available, even if one of them is only implicit.

If p h = 〈 x h , y _ h 〉 is a location addressed from the perspective of the home team with reference to the goal they defend, then p a = 〈 x a , y ¯ a 〉 is the same location from the perspective of the away team and with reference to the goal they attack, if and only if x _h = - x _a and y _ h = – y ¯ a . As depicted in Table 2, changing perspective from one team to the other thus amounts to flipping sides and swapping goals, or 〈 x , ( − y a y h ) 〉 ↦ 〈 − x , ( − y h y a ) 〉 with y _h and y _a the unsigned vertical distances from the goal lines of the home and away team, respectively.

The goal-aligned coordinate system is designed in response to issues caused by standardizing pitch dimensions and using a corner or the center of the pitch as the origin. We therefore outline how some of the system’s properties address these issues. The arguments are organized along the criteria set out in the introduction.

(C1) Representation. In measurement theory, a representation is a mapping between an empirical structure (such as space) and a numerical structure (such as a coordinate system) that preserves a set of relevant relations (such as distances). By using multiples of the same unit of length as coordinates in each dimension, we ensure that dimensions are on the same scale, and thus a direct correspondence between computations in the coordinate system and spatial properties such as distances, areas, and angles.

The inverse problem of locating a point on the pitch that is represented by goal-aligned coordinates only requires knowledge of the association of the two reference points with the actual centers of goals on the pitch. Note that currently used coordinate systems require the same to determine the orientation of the axes.

Because of this straightforward inversion there is no need to reverse what potentially was a non-similarity transformation applied during pitch standardization or to precompute spatial relations before standardizing coordinates. It is interesting to note that ( y _ - y ¯ ) / 2 is the signed distance from the halfway line, so that goal-aligned coordinates can also be seen as a generalization of centered coordinate systems such as TRACAB’s.

(C2) Interpretation. If a player is fouled in location 〈 12 , ( − 15 90 ) 〉 , we can infer that this is within the opposition’s penalty area, because it is 12m to the right of the vertical axis and 15m short of the opposition’s goal line. In other words, we can read off and interpret the exact location directly from the coordinates. This is not the case in systems that involve scaling (let alone non-homogeneous scaling) or varying distances of important field markings from the origin.

For the most important spatial predicates, one of the two vertical coordinates in a goal-aligned coordinate system will allow for a direct interpretation with no transformation necessary. Examples are given in Table 3. Only for locations in touch (horizontally out of bounds), the width of the field is needed as the second contextual parameter.

(C3) Standardization. It goes without saying that coordinates should be assigned consistently over the duration of a match. For analyses covering multiple matches of a team or a player, or comparing locations of different players in different teams and different matches on different fields of play, additional requirements need to be satisfied to ensure commensurability. Standardized dimensions are one way of ensuring that locations appear comparable relative to a corner of a rectangular pitch.

We have seen in the previous section that variable aspect ratios, however, necessitate non-similarity transformations. As a consequence, spatial analyses such as the area around a player when receiving the ball or the distance of defenders from the attackers they mark can be distorted differently on the same pitch (Fig. 3) as well as across pitches (Table 1). This does not happen with goal-aligned coordinates.

Similarly, shot locations from multiple matches are easily aggregated and placed on a common shot map when represented with the attacking goal-aligned coordinate. Contrast this to a coordinate system with the origin at the center, where each shot location needs to be translated because the goal line may be at varying distance from the halfway line.

With goal-aligned coordinate systems, no scaling or other adjustments are needed, except when changing the unit of length from meters to, say, centimeters or yards. Instead, we only select the longitudinal coordinate corresponding to the analytic interest at hand. For defensive actions, for instance, this will generally be the vertical coordinate with reference to the analyzed team’s own goal. For an analysis of counterpressing, however, it may be more appropriate to consider the opposition goal as the reference point.

Given the above properties, two choices in the design of goal-aligned coordinate systems may appear questionable.

For instance, one might argue that flipping the orientation of the vertical axis for coordinates with reference to the opposition goal leads to a higher degree of symmetry and avoids the use of negative coordinates for the most part. From a conceptual point of view, however, this would lead to a mixing of both teams’ perspectives and, more importantly, it would make the comparison of difference vectors (in particular between origin and destination of a pass or run) less straightforward. By retaining the orientation of the vertical axis, results are independent of the choice of vertical coordinate when computing difference vectors or angles.

The main question regards the conceptual redundancy of goal-aligned coordinates. Since y ¯ ≡ y ¯ mod ℓ , it is not apparent why we propose to use two vertical coordinates rather than just one with all computations performed modulo pitch length. The reason is visible in Fig. 4: if the location below the goal line is expressed modulo pitch length, it becomes indistinguishable from the location on the pitch near the other goal line. Locations outside of the pitch are, however, relevant, because a football is not in goal or out of bounds unless it has passed the corresponding line in full. If the ball coordinates refer to its center, the location of the ball may indeed be outside of the pitch with the ball still in play.

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Fig. 4

Goal-aligned coordinate systems are defined by aligning the axes of a Cartesian coordinate system with the line through two given points, G ¯ and G ¯ , the centers of the goal defended and of the goal attacked (left). Each location on the field of play (right) is then represented by one lateral and two complementary longitudinal coordinates, one corresponding to a defensive perspective of a team (distance to give from own goal line) and the other corresponding to an attacking perspective of the same team (distance to go to opposition goal line).

With respect to spatial features, the only variable elements in the specification of a field of play are its dimensions ℓ × w and the unit s of measurement for length. We refer to these three values as contextual data. While it is good practice to include them in any data set, they may also be implied by sport-specific convention, e.g., ℓ=105, w = 68, and s = 1m (one meter).

For storage and transmission of two-dimensional spatial data in goal-aligned coordinates, no additional space is therefore required, provided that contextual data are included or implied. If contextual data is implied, the only essential difference between goal-aligned coordinates and the TRACAB system is the choice of origin. Similar to the systems in StatsBomb and Opta data, we may choose to store player locations with reference to their own goal, and the ball with reference to the home team’s goal.

For in-memory representation, however, it will be more flexible to define views on locations. In an object-oriented programming environment, locations can be encapsulated in a class that references pitch characteristics and offers methods to access the location in any of the four perspectives with respect to either goal as attacked or defended by either team. This is particularly convenient when the analytic interest is in relations between players of opposing teams, such as pressures, pitch control, or the extraction of marking networks. Where relevant for additional context, we may also refer (explicitly or implicitly) to match data that includes the associations of the physical goals with each team in each period.

In writing, locations on a pitch can be specified as usual in a pair of 〈x, y〉-coordinates if the perspective (which team, which goal?) is clear from context. Like most of the action, many tasks in analytics do not require knowledge of pitch dimensions, but are in reference only to one of the goals. Examples include shot maps, expected goals, height of the defensive line, and passes into the final third. Others such as team shape, pass lengths or angles, and line-breaking passes are translation invariant and may not need a reference point at all.