## 3 Determination of the Geometric Model Parameters of SDM
According to the cosine formula, the geometric expression of the cave contour (line a) \(\rho_{k}\) is as follows:  
\[\rho_{k}=\sqrt{{d_{1}}^{2}+{d_{2}}^{2}-2d_{1}d_{2}\cos\beta}. \tag{9}\]  
As can be seen from the sine formula:  
\[\gamma=\arcsin\frac{d_{1}\sin\beta}{\rho_{k}}, \tag{10}\]  
where \(\beta=\frac{\pi}{2}-\alpha\); Formula (10) can be further derived as follows:  
\[\rho_{k}=\sqrt{{d_{1}}^{2}+{d_{2}}^{2}-2d_{1}d_{2}\sin\alpha}. \tag{11}\]  
Figure 8: Geometric relationship of the laser ranging parameters. Figure 7: Schematic diagram depicting the method for determining the cave contour. Similarly, Formula (11) can be further derived as follows:  
\[\gamma=\arcsin\frac{d_{1}\cos\alpha}{\rho_{k}}. \tag{12}\]  
Substituting \(\theta=\alpha-\gamma\) yields the following:  
Figure 7.0  
Therefore, Formula (2) \(\rho_{k}=K(\theta)\) can be derived as follows:  
Figure 7.1  
Finally, the measurement points of section \(Z_{i}\) are plotted in a 3D diagram to form the cave contour (line a). (3) Determination of the surrounding rock stability edge line (line b). Section \(Z_{i}\) in Figure 7 is selected to display the surrounding rock stability edge line (line b), as shown in Figure 7.2 Line b is located inside the rock mass and is not visible; thus, it cannot be measured directly. In order to accurately estimate the thickness of the unstable surrounding rock, a coring survey method is adopted [27; 28; 29; 30; 31]. The same section is selected for the coring survey and laser ranging, but it is more difficult and time-consuming to acquire high-density coring survey points compared to performing laser ranging, and the detection accuracy is limited. Therefore, the core points can be adjusted according to the size of the cave, with no fewer than six core points for each section. Each core point should be drilled into the medium-weathered rock mass to a depth of no less than 2 m [32]; that is, \(d_{p}\geq 2\) m. \(d_{u}\) is the coring depth of the unstable surrounding rock. The \(d_{u}\) depths of each core point are connected to draw a closed stable line b, as shown in Figure 7.3 Then, line b can be expressed as:  
Figure 7.4  
(4) Determination of the filling line (line e). The filling line (line e) of sediments such as gravel, sand, and silt in the cave was measured by the engineering measurement method, which is to measure the distance between the filling surface and the tunnel center point O with a total station. This distance was set to \(y_{4}\), and the value of \(y_{4}\) in Formula (9) was determined. Because \(d_{f}\) may be below or above the point O, \(d_{f}\) satisfies Formula (16). Line e adopts the cartesian coordinate system with the central axis of the tunnel as the origin. It should  
Figure 7.5 Determination of the surrounding rock stability edge line (line b). be noted that, when determining the coring survey of line b, the thickness \(d_{f}\) of the filling material can be determined simultaneously, as shown in Figure 7.6 Then, line e can be expressed as follows:  
Figure 7.7  
(5) Determination of the water level (line f). The water level (line f) of the cave is spatiotemporally dependent, such that the water level should be a change interval. The variation range of the water level is obtained by observing the historical trace of the water level on the karst cave wall. The value of \(y_{3}\) is the maximum over the years, and the water depth is set as \(d_{w}\), as shown in Figure 7.8 Again, because \(d_{w}\) may below or above point O, \(d_{w}\) satisfies Formula (18). Then, the water level line can be expressed as follows:  
Figure 7.9  
(6) Geometric drawing of SDM. After the geometric parameters of the SDL are determined, the spatial and structural diagram of the karst cave and tunnel can be drawn, as illustrated in Figure 8.0 From this structural diagram, the spatial position relation between the cave and the tunnel is immediately clear, thus realizing the realistic transformation of the SDM.