## 2 Methodology
### Visibility Analysis
#### 2.1.1 Parameter Optimization  
The vertical angle takes into account the relative distance and relative height between tourists and the landscape. Without considering the terrain where tourists are located, the impacts of the relative distance and relative height on the vertical angle are shown in Figure 1. By simplifying and amplifying the slope and height of the terrain where the landscape is located within a two-dimensional triangular plane, as the vertical height decreases and the distance between the landscape and the tourists increases, the vertical angle becomes smaller, visibility weakens and the visual experience of sightseeing becomes worse. In contrast, the larger the vertical angle, the better the visibility and the sightseeing experience. The calculation formula for the vertical angle is:  
\[\alpha=\arctan(\frac{\Delta\mathrm{H}}{L}) \tag{1}\]  
In Figure 1, point A represents the tourist, and point B represents the landscape. Using the elevation of point A where the tourist is located as the reference line, \(\alpha\) represents the vertical angle of point A, \(L\) represents the actual distance between the two points and \(\Delta\mathrm{H}\) represents the elevation difference between the two points. However, the above ideal model does not exist because the slope and height of the terrain where the tourists are located also affect the accuracy of the vertical angle calculation. Therefore, we enlarge and simplify both the slope and height of the terrain where the tourists A and the landscape B are located within a two-dimensional triangular plane, respectively (Figure 2). Figure 1: The effects of relative distance and elevation difference on vertical angle without considering the terrain where tourists are located. Where point A represents the tourist, point B represents the landscape, h\({}_{1}\) and h\({}_{2}\) represent the elevation where the tourists and the landscape are located, respectively, L represents the relative distance between the two and \(\beta_{1}\) and \(\beta_{2}\) represent the terrain slope where they are located, respectively. Therefore, after considering the terrain where the tourists are located, the calculation formula for the vertical angle is:  
\[\alpha=\arccos(\frac{x_{4}^{2}+x_{3}^{2}-x_{2}^{2}}{2x_{4}x_{3}}) \tag{2}\]  
Because:  
\[x_{1}=\frac{\text{h}_{1}}{\sin\beta_{1}} \tag{3}\]  
\[x_{2}=\frac{\text{h}_{2}}{\sin\beta_{2}} \tag{4}\]  
\[x_{3}=\sqrt{(x_{1}^{2}+L^{2}-2x_{1}L\cos(180^{\circ}-\beta_{1}))} \tag{5}\]  
\[x_{4}=\sqrt{(x_{2}^{2}+x_{3}^{2}-2x_{2}x_{3}\cos\theta)} \tag{6}\]  
\[\gamma=\text{arc}\;\cos(\frac{L^{2}+x_{3}^{2}-x_{1}^{2}}{2 Lx_{3}}) \tag{7}\]  
\[\theta=180^{\circ}-\gamma-\beta_{2} \tag{8}\]  
We derive Equation (9) from Equations (3) and (5):  
\[\alpha=\arctan(\frac{\Delta\mathrm{H}}{L}) \tag{1}\]0  
Then, Equation (10) can be derived from Equations (7)-(9):  
\[\alpha=\arctan(\frac{\Delta\mathrm{H}}{L}) \tag{1}\]1  
Equation (11) is derived from Equations (4), (6), (9) and (10):  
\[\alpha=\arctan(\frac{\Delta\mathrm{H}}{L}) \tag{1}\]2  
Finally, the vertical angle \(\alpha\) is calculated by combining Equations (2), (4), (9) and (11). The larger the vertical angle, the better the visual experience for the tourists. \[\alpha=\arctan(\frac{\Delta\mathrm{H}}{L}) \tag{1}\]3 Vertical angle calculation schematic diagram after considering the terrain where tourists are located. Evidently, the larger the slope of the landscape surface relative to the tourists' perspective, the larger the visible area and the likelihood of the landscape being noticed, and the more comfortable the visual sightseeing experience. Therefore, the projected area of the landscape surface along the line of sight can be used to describe the impact of the relative slope on the tourists' visual experience (denoted by sensitivity S), as shown in Figure 3 (level sight as an example). For level or upward sight, the relative slope is the actual terrain slope. For downward sight, the relative slope is the terrain slope minus \(90^{\circ}\).