# Two-Lane Highways Crest Curve Design. The Case Study of Italian Guidelines
## 3 Analytical Derivation of Crest Curve Minimum Radius  
The minimum radius of crest curves based on passing sight distance (PSD) criteria generally is satisfactory from the standpoint of safety, comfort, and appearance [15,22]. In this section, the detailed derivation process for the analytical relationship between passing sight distance (PSD), driver eye height h\({}_{1}\), object (i.e., opposing vehicle) height h\({}_{2}\), and the necessary parabolic vertical crest curve radius R\({}_{v}\) is presented.  
Let the left-hand side road grade be i\({}_{1}\), and right-hand side road grade i\({}_{2}\), the total grade change \(\Delta\)i = i\({}_{2}-\)i\({}_{1}<0\) (assuming i\({}_{2}<\)i\({}_{1}\)), and L the crest vertical curve length. The origin of the coordinates is selected at point O (beginning of the vertical curve) (Figure 3). If the vertical curve follows a parabolic shape, the curve equation is:  
\[\text{y}=\text{ax}^{2}+\text{bx} \tag{3}\]  
The coefficients a and b can be obtained by the following boundary conditions:  
\[\left(\frac{\text{d}\text{y}}{\text{d}\text{x}}\right)_{\text{x}=0}=\text{b} \ \ \text{therefore}\ \ \ b=\text{i}_{1} \tag{4}\]  
Figure 3: Parameters considered in determining the crest vertical curve radius.  
Figure 2: Passing sight distance (PSD) models comparison.  
\[\frac{\mathrm{d}^{2}\mathrm{y}}{\mathrm{dx}^{2}}=2\mathrm{a}\ \ \ \mathrm{therefore}\ \ \ \mathrm{a}=\frac{\Delta\mathrm{i}}{2\mathrm{L}} \tag{5}\]  
The vertical curve equation \(\mathrm{y}=\mathrm{ax}^{2}+\mathrm{bx}\) can be rewritten as Equation (6)  
\[\mathrm{y}=\frac{\Delta\mathrm{i}}{2\mathrm{L}}\mathrm{x}^{2}+\mathrm{i}_{1} \mathrm{x} \tag{6}\]  
the coordinates in the summit of the parabola (point A of Figure 3) are:  
\[\mathrm{x}_{\mathrm{A}}=-\frac{\mathrm{i}_{1}}{\Delta\mathrm{i}}\mathrm{L} \tag{7}\]  
\[\mathrm{y}_{\mathrm{A}}=-\frac{\mathrm{i}_{1}^{2}}{2\Delta\mathrm{i}}\mathrm{L} \tag{8}\]  
the relationship of the curvature is:  
\[\frac{1}{\mathrm{R}}=\frac{\frac{\Delta\mathrm{i}}{\mathrm{L}}}{\left[1+\left( \frac{\Delta\mathrm{i}}{\mathrm{L}}\mathrm{x}+\mathrm{i}_{1}\right)\right]^{ 2}} \tag{9}\]  
Equation (9) allows to calculate the radius of the osculating circle in the points O, A, and B:  
\[\mathrm{R}_{\mathrm{o}}=\frac{\left[1+\mathrm{i}_{1}^{2}\right]^{\frac{3}{2} }}{\Delta\mathrm{i}}\mathrm{L} \tag{10}\]  
\[\mathrm{R}_{\mathrm{B}}=\frac{\left[1+\left(\mathrm{i}_{1}+\Delta\mathrm{i} \right)^{2}\right]^{\frac{3}{2}}}{\Delta\mathrm{i}}\mathrm{L} \tag{11}\]  
\[\mathrm{R}_{\mathrm{A}}=\frac{\mathrm{L}}{\Delta\mathrm{i}} \tag{12}\]  
The minimum radius of curvature \(\mathrm{R}_{\mathrm{v}}\) is located at point A, therefore we can assume \(\mathrm{R}_{\mathrm{v}}=\mathrm{R}_{\mathrm{A}}\). Let \(\mathrm{D}=\mathrm{PSD}\) (cfr. Figure 4). The relationship between the minimum radius of curvature \(\mathrm{R}_{\mathrm{v}}\), the passing sight distance \(\mathrm{D}\), the driver eye height \(\mathrm{h}_{1}\), object (i.e., opposing vehicle) height \(\mathrm{h}_{2}\), and the total grade change \(\Delta\mathrm{i}\) can be derived using the geometric schemes, as depicted in Figure 4. More specifically, Figure 4 illustrates the influence of vertical curvature on visibility. Two conditions must be considered [1; 9; 22; 31]:  
1. Passing sight distance is shorter than vertical curve length (\(\mathrm{D}<\mathrm{L}\));
2. Passing sight distance is longer than vertical curve length (\(\mathrm{D}>\mathrm{L}\)).  
Figure 4: Crest curve radius derivation (case (**a**): \(\mathrm{D}<\mathrm{L}\); case (**b**): \(\mathrm{D}>\mathrm{L}\)).