## 4 Results and Discussion
### Lateral Displacement for Lane-Changing Overtaking  
According to the overtaking process with lane-changing behavior, the absolute displacement of the overtaking vehicle is much larger than that of the overtaken vehicle, and there is no need to perform much comparison with lane-keeping overtaking. Taking 1 m intervals, the overtaking trajectories in Class III are arranged in ascending order at each cross-section to obtain the typical percentile trajectory curves of consecutive sections, as shown in Figure 10(a). Similarly, the typical percentile curves of lateral speed in Figure 10(b) can be obtained. It is found that the 50 th percentile trajectory before overtaking is already in the lane-changing process at X = \(-\)100 m, while the 50 th percentile trajectory after overtaking has already finished lane changing at about X = 90 m. Combined with the lateral speed curves, it can be seen that the overtaking vehicle tends to prepare in advance before overtaking and completes overtaking over  
Figure 10: The relationship between initial lateral spacing and relative displacement of vehicles. (**a**) Before overtaking; (**b**) After overtaking. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Stage** & **Coefficient** & **Value** & **Standard Error** & **t** & \(\mathbf{p}\) & \(\mathbf{R}^{2}\) \\ \hline \multirow{2}{*}{Before overtaking} & Linear term & \(-\)0.530 & 0.017 & \(-\)31.50 & \(<\)0.001 & \multirow{2}{*}{0.356} \\  & Constant term & 2.250 & 0.069 & 32.51 & \(<\)0.001 & \\ \hline \multirow{2}{*}{After overtaking} & Linear term & \(-\)0.468 & 0.024 & \(-\)19.85 & \(<\)0.001 & \multirow{2}{*}{0.211} \\  & Constant term & 1.757 & 0.099 & 17.81 & \(<\)0.001 & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Linear fitting results of initial lateral distance and relative displacement. a longer distance with a lower lateral speed; on the contrary, the lane-merging process after overtaking is completed in a shorter distance with a higher lateral speed. The safe following distance of the freeway is generally greater than 100 m, and changing lanes as early as possible before overtaking is conducive to obtaining better visual conditions and sufficient reaction time. It should be noted that most of the merging behaviors are completed within the safe following distance. In this case, the overtaken vehicle cannot accelerate sharply to avoid the potential risk of rear-end collision. By analyzing the trajectory and lateral speed curves at different quantile values, it is evident that the degree of curve distribution dispersion varies across sections. The magnitude of the standard deviation can be used as an indicator to assess this dispersion, as illustrated in Figure 11c. We found that the standard deviation takes the minimum when the vehicles are parallel. In other words, the driving behaviors are convergent. When the driving parameters are distributed in a narrow interval, the design parameters of the road can be adjusted in a smaller range to take care of the behavioral habits of most drivers. Therefore, when the lateral distance of vehicles is taken as the control factor for lane width design, it is more representative to choose the lateral distance at the moment of vehicle parallelism. Although the trajectories before and after overtaking come from different samples, the typical percentile values of the trajectories at X = 0 are close and the trend is also smooth. Furthermore, existing studies have confirmed that \"50 th percentile drivers\" and \"85 th percentile drivers\" exist in the real world. In other words, drivers who are highly close to the trend of a typical percentile curve can always be found in the sample. Given this, a relatively complete overtaking trajectory can be obtained by stitching together the trajectory curves for the same percentile values before and after the overtaking. It is noted that 85% of the overtaking vehicles can complete overtaking beyond the 15 th percentile trajectory curve.