## 2 Materials and Methods
### Determination of the Overlapping  
Starting with the first main part pass, they were sequentially analyzed in terms of the lateral overlapping. The overlaps between the cultivated tracks of an arbitrary pass and the five chronologically subsequent passes were identified to ensure that no overlap was ignored. The overlap identification was done by determining for each comparison the intersection polygons and calculating their area. For this, the MATLAB functions \"polybool (intersection)\" and \"polyarea\" were used. The same process was performed on the headland passes. However, the overlaps between each cultivated track of a headland pass and all its chronologically subsequent ones were identified.  
The overlaps between the main part passes and the headland passes were determined separately. Technically, the analysis procedure was the same as for the lateral overlaps. In this case, however, for every main part pass of a field, the overlaps with all the headland passes were identified. Naturally, it was possible that there was concurrently a lateral overlap and a headland overlap at locations, where main passes reach into the headland. They were respected separately in the calculations. Figure 10  
Figure 9: Schematic illustration of the modeling of the cultivated area for three exemplary passes.  
illustrates the conception of the boundary and overlaps in an idealized way. The main part passes are running in an east-west direction and the headland passes are running in a north-south direction.  
To enable a differentiated analysis of the overlapping within the field, different intermediate figures were determined. For the \(A_{overlap,lat}\), the area of all lateral overlaps of the field was added up. The total headland overlapping for every field is expressed by \(A_{overlap,head}\), which is the summed area of all overlapping polygons resulting from the intersections between the headlands and the main part. The first visual assessments of the modeled cultivated tracks uncovered noticeable lateral overlaps alternating with uncultivated lateral gaps due to the receiver error. Thus, the overlapping was corrected with the area of the gaps (\(A_{gaps}\)), which was calculated as follows:  
\[A_{gaps}=A_{boundary}-\left(A_{total}-\left(A_{overlap,lat}+A_{overlap, head}\right)\right) \tag{2}\]  
A dedicated analysis of the lateral overlapping was enabled by calculating the \(A_{overlap,lat,perc}\), which indicates the relation between the area of the corrected lateral overlapping and the total cultivated area within a field:  
\[A_{overlap,lat,perc}= \left(A_{overlap,lat}-A_{gaps}\right)/A_{total}\cdot 100\% \tag{3}\]  
To enable a distinct assessment of the headland overlapping, the \(A_{overlap,had,perc}\) was determined as an indicator for the relation between the summed area of headland overlaps and the total cultivated area of a field:  
\[A_{overlap,head,perc}= A_{overlap,head}/A_{total}\cdot 100\% \tag{4}\]  
\(A_{overlap,perc}\) is considered the adequate indicator for the total overlapping within a field. It was calculated as follows:  
\[A_{overlap,perc}= \left(\left(A_{overlap,lat}-A_{gaps}\right)+A_{overlap,head} \right)/A_{total}\cdot 100\% \tag{5}\]  
Thus, it is the sum of \(A_{overlap,lat,perc}\) and \(A_{overlap,head,perc}\).  
To compare the \(A_{overlap,perc}\) with a more common indicator, which indicates the overlapping percentage within a field, the \(A_{overlap,perc,ref}\) was calculated additionally:  
\[A_{overlap,perc,ref}=\left(A_{total}-A_{boundary}\right)/A_{boundary}\cdot 1 00\% \tag{6}\]  
The \(A_{overlap,perc,ref}\) was inspired by how [PERSON] et al. [5] set the total cultivated area in relation to the area of the field boundary. This is a common way to express the overlapping by means of commercially available PF terminals.  
Figure 10: Schematic illustration of the concept of the boundary and overlaps.