# Three-Dimensional Stopping Sight Distance Calculation Method under High Slope Restraint
## 3 Calculation of Horizontal S.S.D. In this paper, based on the horizontal and vertical combination, the three-dimensional sight distance is studied, and the horizontal S.S.D. is calculated by the plane design drawing. It has been assumed that the car is driving in the center of the lane. In AutoCAD (AutoDesk Inc., San Rafael, CA, USA), the road centerline can be shifted the distance of halfway width of the lane to obtain the driving trajectory. In the transition curve section of the curve after the offset, take a point S\({}_{0}\) as the line of sight calculation point, such as Figure 2, where \(Q\) is the tangent point between line of sight and slope. Extracting the \(S_{0}\) coordinates (\(x_{0}\), \(y_{0}\)) and the center \(O\) coordinate (\(x_{0}\)', \(y_{0}\)'), the \(S_{0}O\) distance can be obtained by the Equation (1):  
\[|S_{0}O|=\sqrt{\left(x_{0}^{\prime}-x_{0}\right)^{2}+\left(y_{0}^{\prime}-y_{0 }\right)^{2}} \tag{1}\]  
Since the driving trajectory line is obtained by shifting the road center line to the inner side of the curve in the above calculation, the actual length of the driving trajectory line is slightly shorter than the length of the road center line. According to Equation (2), the length reduction of the offset curve relative to the original curve can be calculated, so the shortened easement curve length can be obtained. \[{l_{s}}^{\prime}=l_{0}-\beta\times D=l_{0}-l_{0}^{2}/\left(2R\times ls\right) \times D \tag{2}\]  
In the formula:  
\({l_{s}}^{\prime}\) is the distance from the SC' (point of spiral to curve) to the TS' (point of tangent to spiral) of the curve after shifting inward, the solution here is the full length of the transition curve;  
\({l_{s}}\) is the length of the original transition curve;  
\({l_{0}}\) is the distance from any point on the transition curve of the original road center line to the starting point TS of the transition curve;  
\(\beta\) is the transition curve angle;  
\(D\) is an internal shift value;  
\(R\) is the radius of the circular curve. Use the tangent offset method (Figure 3) to calculate the distance from the point \(S_{0}\) to the offset TS' point:  
\[\left\{ \begin{array}{c}x=l-\frac{\beta}{40R^{2}L_{s}^{2}}\\ y=\frac{\beta}{6 RL_{s}}-\frac{\beta^{2}}{336R^{3}L_{s}^{3}}\end{array} \right. \tag{3}\]  
Figure 2: Schematic diagram of calculation. In the formula:  
\(x\), \(y\) are the coordinates of the points sought;  
\(R\) is the radius of the curve;  
\(l\) is the distance from the point of the request to the point TS;  
\(L_{s}\) is the length of the transition curve. However, it is worth noting that Equation (3) is the coordinate system established with TS point of the offset curve as the origin of coordinates and the tangent direction of TS point as the \(X\)-axis. Therefore, coordinate system transformation of the coordinate system of S\({}_{0}\) point is required before using this formula. The transformation formula is as follows:  
\[\left\{ \begin{array}{l}X=X_{TS}+x\times cosA\pm y\times sinA\\ Y=Y_{TS}+x\times sinA\pm y\times cosA\end{array} \right. \tag{4}\]  
In the formula:  
\(A\) is the azimuth of the front straight line;  
\(x\) is \(X_{s0}-X_{TS}\);  
\(y\) is \(Y_{s0}-Y_{TS}\). Substituting the coordinates of the converted point \(S_{0}\) into the Equation (3) can obtain the length from the point \(S_{0}\) to the point TS' of the post-offset curve. Assume that the driver drives to \(S_{0}\) at this time, and his line of sight is blocked by the high slope at point \(Q\). If the next moment, the driver finds that there is an obstacle at the point \(D\) of the line of sight extension (on the driving trajectory). If the driver immediately takes braking measures at this time, the available braking distance is \(|S_{0}D|\), and \(|S_{0}D|\) is the S.S.D. value on the plane, as shown in Figure 4.  
Figure 4: Schematic diagram of horizontal stopping sight distance (S.S.D.) calculation. Figure 3: Schematic diagram of the tangent offset method.