# Two-Lane Highways Crest Curve Design. The Case Study of Italian Guidelines
## 3 Analytical Derivation of Crest Curve Minimum Radius
### Case in Which \(D>L\)  
Consider the case in which the passing sight distance is longer than the vertical curve length (D = PSD \(>\) L). Using the geometric parameters in Figure 4, it results in [22]:  
\[\mathrm{h_{1}}=\mathrm{a}\cdot\mathrm{p}^{2}+2\cdot\mathrm{a}\cdot\mathrm{p} \cdot\mathrm{m_{1}} \tag{18}\]  
\[\mathrm{h_{2}}=\mathrm{a}\cdot\mathrm{q}^{2}+2\cdot\mathrm{a}\cdot\mathrm{q} \cdot\mathrm{m_{2}} \tag{19}\]  
\[\mathrm{m_{1}}=\frac{\mathrm{h_{1}}}{2\cdot\mathrm{a}\cdot\mathrm{p}}-\frac{ \mathrm{p}}{2} \tag{20}\]  
\[\mathrm{m_{2}}=\frac{\mathrm{h_{2}}}{2\cdot\mathrm{a}\cdot\mathrm{q}}-\frac{ \mathrm{q}}{2} \tag{21}\]  
\[\mathrm{D}=\mathrm{m_{1}}+\mathrm{L}+\mathrm{m_{2}}=\frac{\mathrm{h_{1}}}{2 \cdot\mathrm{a}\cdot\mathrm{p}}+\frac{\mathrm{h_{2}}}{2\cdot\mathrm{a}\cdot \left(\mathrm{L}-\mathrm{p}\right)}+\frac{\mathrm{L}}{2} \tag{22}\]  
The minimum value of the sight distance can be obtained deriving the Equation (22) and by the following condition:  
\[\frac{\mathrm{d}\mathrm{D}}{\mathrm{d}\mathrm{p}}=\frac{-\mathrm{h_{1}}}{2 \cdot\mathrm{a}\cdot\mathrm{p}^{2}}+\frac{\mathrm{h_{2}}}{2\cdot\mathrm{a} \cdot\left(\mathrm{L}-\mathrm{p}\right)^{2}}=0 \tag{23}\]  
then  
\[\mathrm{p}=\frac{\mathrm{L}}{1+\sqrt{\frac{\mathrm{h_{2}}}{\mathrm{h_{1}}}}} \tag{24}\]  
\[\mathrm{q}=\mathrm{L}-\mathrm{p}=\mathrm{p}\cdot\sqrt{\frac{\mathrm{h_{2}}}{ \mathrm{h_{1}}}} \tag{25}\]  
plug Equations (24) and (25) into Equation (22), after minor rearrangement, it results in:  
\[\mathrm{D}=\frac{\mathrm{L}}{2}+\frac{\mathrm{h_{1}}+\mathrm{h_{2}}+2\cdot \sqrt{\mathrm{h_{1}}\cdot\mathrm{h_{2}}}}{\Delta\mathrm{i}} \tag{26}\]  
\[\mathrm{L}=2\left(\mathrm{D}-\frac{\mathrm{h_{1}}+\mathrm{h_{2}}+2\cdot\sqrt{ \mathrm{h_{1}}\cdot\mathrm{h_{2}}}}{\Delta\mathrm{i}}\right) \tag{27}\]\[\mathrm{h_{2}}=\mathrm{a}\cdot\mathrm{q}^{2}+2\cdot\mathrm{a}\cdot\mathrm{q} \cdot\mathrm{m_{2}} \tag{19}\]0  
According to Equation (28), the minimum radius of curvature \(\mathrm{R_{V}}\) (point A of Figure 3) is a function of the total grade change \(\Delta\mathrm{i}\). Passing Sight Distance, Opposing Vehicle Height \(\mathrm{h_{2}}\), and Eye Height \(\mathrm{h_{1}}\)  
An object (i.e., opposing vehicle) height of \(\mathrm{h_{2}}=1.10\) m is adopted by the Italian Guidelines for the Design of Road Infrastructures [10] for the calculation of passing sight distance PSD. Such value appears not to be in compliance with the average height of the current passenger car population. In fact, over the past few decades, sales of sport utility vehicles (SUVs) have increased. At the same time, even utility cars have undergone increases in overall size and, in particular, in height. For this reason, a specific analysis was carried out concerning the Italian passenger car population until 2006, coinciding with the year of publication of the latest Italian Guidelines for the Design of Road Intersections [11]. The research was carried out using the Italian car population data updated to 31 December 2006, registered by the Italian Automobile Club (ACI) and by the Public Vehicle Registration Office (PRA) [32]. On that date, the vehicles circulating in Italy were 35,297,282. The study only considered car models with more than 20,000 registered vehicles. Specifically, 115 vehicle models belonging to different brands were taken into consideration, for a total of over 9 million vehicles. For each of the 115 models considered, the overall height was deduced. Table 3 shows the summary data. In the study, the average vehicle heights \(\mathrm{h_{2(m)}}\) and the value corresponding to the 15 th percentile of the height distribution \(\mathrm{h_{2(15)}}\) were estimated, obtaining, respectively, the values \(\mathrm{h_{2(m)}}=1.48\) m and \(\mathrm{h_{2(15)}}=1.39\) m.  
As shown in the histogram of Figure 5, the largest number of registered vehicles (4,151,428) falls in the height range 1.40-1.44 m, follow the intervals 1.45-1.49 m and 1.50-1.54 m, with 1,449,211 and 1,100,024 vehicles, respectively.