## 4 Results Comparative Presentation and Discussion  
In this section, results from the models developed in the preceding one are presented and compared. The objective was to solve for the maximum highway speed without slippage between vehicle and pavement, which constitutes a reasonable estimate of a motor road's maximum safe operational speed. They reflect the broad versatility of classical dynamics applied to the problem. The analytical results obtained can be condensed in the following table (Table 1). The visible points of interest are the following:  
(1) Mathematically (i.e., symbolically) comparing equations Equations (2), (13), and (14), it is noticeable that a worldwide used model in regulations is analytically similar/equivalent to the models represented by Equations (13) and (14), particularly the latter. That is, the model expressed in Equation (2) is a slight re-parametrization of Equation (13), mainly in order to consider friction as a demand parameter. Therefore, one of the most used models regarding regulations on highway speeds is a 2D native one applied to a 3D organic situation, as explained in the previous section. (2) As expected, the normal-tangential coordinate motion description (Figure 1, Equation (7)) and the unrotated cylindrical-polar system coordinate motion description (Figure 5, Equation (16)) models are completely analytically equivalent. A non-analytical difference, but conceptually substantial, is related to the nature of the motion curvature parameter used. In the former, the circular trajectory curvature radius (\(R\)) is in a plane parallel to that used for super-elevation measurement. Notice that this definition is only possible by adding a new coordinate normal to the \"n & t\" original set, _ad-hoc_ to the problem; that is, an _ad-hoc_ renderization of a 2D situation into a 3D one is required. Now, in the latter, the coordinate \"added\" is one of the natural coordinates of the model, the radial one (\(r\)). The plane of motion considered is also parallel to that used for super-elevation measurement. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Geometric Model** & **Curvature Parameter** & **Maximum Speed** \\ \hline “n \& t”/unrotated cylindrical-polar system (Equation (7)/Equation (16)) & \(R\) & \(v=\sqrt{\frac{gR(sim+\mu cos\theta)}{cos\theta-isim}}\) \\ Rotated cylindrical-polar system (Equation (13)) & \(r\) & \(v=\sqrt{gr(sim+\mu cos\theta)}\) \\ Rotated cylindrical-polar system (Equation (14)) & \(R\) & \(v=\sqrt{gR(\mu+tan\theta)}\) \\ Spherical system (Equation (21)) & \(R\) & \(v=\sqrt{\frac{grect(\mu+tan\theta)}{1-\mu sin\theta}}\) \\ Spherical system (Equation (22)) & \(r\) & \(v=\sqrt{\frac{gR(\mu+tan\theta)}{1-\mu tan\theta}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of results for safe highway operating speeds \(v\) for different models. Geometrically-related parameters used are motion curvature radius (\(R\)) or intrinsic radial (\(r\)) coordinate to the specific model, the friction coefficient between vehicle and pavement (\(\mu\)), and highway super-elevation (\(\theta\)). \(g\) is the gravitational acceleration’s magnitude considered constant for the analysis. (3) In Equation (12), the acceleration does not have components in addition to the radial one because the cylindrical-polar coordinate system as a whole rotated with respect to the horizontal plane is considered, an angle amounting to the highway super-elevation \(\theta\). In such a system, the motion studied does not take place on the horizontal Cartesian plane \(\hat{x}-\hat{y}-\hat{z}\) but in the rotated plane \(\hat{r}-\hat{\phi}-\hat{z}\) described by the cylindrical-polar coordinate system, as shown in Figure 4. The radial direction \(\hat{r}\) is then perpendicular to the studied motion's tangential direction. In other words, in the instant represented in Figure 4, \(\hat{\phi}\) enters the scene.