## 2 Materials and Methods
### Modeling of the Representative Plow Points  
As a preparatory step, relevant sizes were taken from the real tractor-implement combination, whereby the working width was set to its maximum. In Figure 8, the modeling of the representative plow points is schematically illustrated and the relevant dimensions are indicated. Figure 8a shows the plow rotated on the left rotating position, while Figure 8b shows the plow rotated right. The figure is kept abstract because it is supposed to serve only as a schematic illustration. To enable a better overview, only the shares that are operating in each case are presented. A distinction of the two rotating positions is important because they represent two mirror-inverted arrangements of the working tools, which has an important effect on the reconstruction of the cultivated track.  
Figure 7: Schematic operating principle of the algorithm for the distinction between main part passes and headland passes.  
For every rotating position separately, two points were defined that represented the lateral edges of the working width (points B and \(\Gamma\)). As can be seen in Figure 8a,b, the representative flow points are formed by first drawing a line parallel to the outer edge of the first and last share and crossing it with a perpendicular line, which is touching the front top of the shares. Then, this intersection point is projected vertically to the height of the antenna (Point A) to get the representative flow point. For the subsequent calculations, the left and right plow vectors pointing from the antenna to the representative points of the plow were created. Then, the \(\varphi\) angle (\(\varphi_{\text{L}}\) and \(\varphi_{\text{R}}\) for both rotating positions) and the \(\theta\) angle (\(\theta_{\text{L}}\) and \(\theta_{\text{R}}\) for both rotating positions) were determined. They represent the angle between the forward direction and the left and right plow vectors. Furthermore, the length of the left and right plow vectors for both rotating positions (AB\({}_{\text{L}}\) and AB\({}_{\text{R}}\), and AT\({}_{\text{L}}\) and AT\({}_{\text{R}}\), respectively) was determined. The values for these dimensions, which indicate the position of the representative flow points relative to the antenna position, are presented in Table 2.  
Several adjustment options of the plow have a substantial effect on its effective working width. As the logged data could not deliver exact information about the work state, assumptions were made concerning the main parameters. The working width was assumed to always be set to its maximum during the data acquisition. In fact, this was plausible, as the tractor had a relatively high power compared to the plow's size. Concerning the rotation status, a simplified scheme was developed and applied to simulate this special characteristic. First, a distinction of passes having the same direction as the very first pass of a field part and passes running in the opposite direction was performed. For this, the angle between the vector of the very first and all other passes of a field part was determined using Equation (1). If the angle was between 0\({}^{\circ}\) and 45\({}^{\circ}\), the pass was declared as running in the same direction as the very first pass. Then, the plow was assumed to be rotated right according to the definition in Figure 8. If it was between 135\({}^{\circ}\) and 180\({}^{\circ}\), it was assumed to belong to a pass running in the opposite direction and, thus, rotated left.