## 5 Parametric Sweep: Visualization and Analysis  
In order to compare models among them, an assessment of the parameters involved must be conducted first. There are four of them: \(r\), \(R\), \(\theta\), and \(\mu\). Regarding the last one, the friction coefficient between pavement and vehicle (\(\mu\)) is an experimentally obtained parameter. Naturally, it has a physical nature, though not much of a geometrophysical one in the sense implied here. The remaining parameters are fully geometrophysical since they arise from analyzing and applying different conceptual/geometrophysical models over the problem of interest discussed previously. Therefore, the parametric set up is shown in Table 2, according to Mexico's governmental agency in charge of highway design regulations [18] and experimental procedures in Table 3, fully described in [27]. Figures 7-12 show the graphical results of the parametric analysis proposed. Before briefly discussing them, analytical calculations were developed on default MATLAB R2022b computational machine precision, retaining three significant digits due to friction coefficient experimental values (significant figures) in [27]. On the other hand, as expressed in the figure captions, the calculations for percentage relative errors took as reference the rotated cylindrical/radial coordinate model in Equations (13) and (14) due to its analytical/conceptual proximity to the model in Equation (2), as explained in previous sections: the model in Equation (2) is a reasonable numerical approximation of the model in Equation (1). Concurrently, the models in Equations (13) and (14) are equivalent among them (when switching the curvature parame \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Parametric Range** & **Parameter Mean** \\ \hline Curvature radius & \(19.1\leq R\leq 416.7\left[m\right]\) & \(\sim\)218 \(\left[m\right]\) \\ Super-elevation & \(0^{\circ}\leq\theta\leq 5.71^{\circ}\) & \(\sim\)3\({}^{\circ}\) \\ Maximum speed & \(30\leq v\leq 110\left[km/h\right]\) & \(\sim\)70 \(\left[km/h\right]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mexico’s governmental sanctioned highway design parameters. Extracted from [18]. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Pavement Particular Conditions** & **Parametric Range** & **Parameter Mean** & **Mean Particle Diameter** \\ \hline Iced, no sand & 0.10 \(\leq\mu_{s}\leq\) 0.23 & \(\sim\)0.1650 & Not applicable. \\ Dry, sandy & 0.56 \(\leq\mu_{s}\leq\) 0.66 & \(\sim\)0.61 & \(\sim\)176 \(\left[\mu m\right]\) \\ Dry, sandy & 0.71 \(\leq\mu_{s}\leq\) 0.79 & \(\sim\)0.75 & \(\sim\)10\({}^{3}\) \(\left[\mu m\right]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimentally obtained friction coefficients under different conditions between tire rubber and pavement. Contact/interface area at temperatures including \(-10\) [\({}^{\circ}\)C] \(\leq T\leq 0\) [\({}^{\circ}\)C]. Extracted from [27]. ter considered, i.e., \(r\to R\)) and also a straight forward reparametrization of the model in Equation (2). Figure 8: Maximum velocity vs. super-elevation angle. \(0.56\lesssim\mu\lesssim 0.66\), \(\rho=218\ [m]\). \(\rho\) is the curvature parameter used, whether \(r\) or \(R\). Pavement conditions: dry, sandy. Sand particle mean diameter \(\sim\)176 \([\mu m]\). In this figure, the percentage relative errors between models, having as a reference the rotated cylindrical/curvature radius (that is presented in Equation (14)), vary between \(0\leq\%E\leq 3.94\). Figure 7: Maximum velocity vs. super-elevation angle. \(18\lesssim\rho\lesssim 418\ [m]\), \(\mu=0.61\). \(\rho\) is the curvature parameter used, whether \(r\) or \(R\). Pavement conditions: dry, sandy. Sand particle mean diameter \(\sim\)176 \([\mu m]\). In this figure, the percentage relative errors between models, having as a reference the rotated cylindrical/curvature radius (that is presented in Equation (14)), vary between \(0\leq\%E\leq 3.65\)  
Figure 10: Maximum velocity vs. super-elevation angle.