# A Global Continuous Multi-Scale Terrain Contour Simplification Method Based on the Level Set
## 4 Discussion  
### Effect of CCS's Mesh Parameters on Similarity  
To examine the effect of CCS's mesh parameters on the similarities among simplified contour lines, we used different mesh parameters for CCS's tessellation. First, as shown in Figure 12, the contour with 86 points was taken as the initial contour and was labeled C\({}_{1}\). By using the DP algorithm, C\({}_{1}\) was simplified into a target contour labeled C\({}_{2}\) with only the starting and ending points of C\({}_{1}\) (representing the situation of extreme simplification), whereas C\({}_{1}\) was simplified into a target contour labeled C\({}_{35}\) with 35 points of C\({}_{1}\) (representing the situation of middle simplification).  
We then built CCS-2 by using C\({}_{1}\) (initial contour) and C\({}_{2}\) (target contour) and built CCS-35 by using C\({}_{1}\) (initial contour) and C\({}_{35}\) (target contour). Both CCS-35 and CCS-2 were established under different mesh parameters (i.e., 0.01, 0.008, 0.006, 0.004, 0.002, and 0.001). Figure 13 shows CCS-35 and CCS-2 under different mesh parameters.  
Figure 12: Different contours with different points. (**a**) The initial contour (C\({}_{1}\)) with 86 points. (**b**) The simplified contour (C\({}_{35}\)) with 35 points. (**c**) The simplified contour (C\({}_{2}\)) with 2 points.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Std (m *)** & **Mean (m)** & **Max (m)** \\ \hline The DP algorithm & 208.297 & 451.011 & 778.567 \\ The WM algorithm & 208.246 & 451.158 & 778.640 \\
**CCSM** & **208.199** & **451.053** & **778.478** \\ \hline (*m = meter). & & & \\ \end{tabular}
\end{table}
Table 3: Numerical evaluation of the positional errors between the initial and simplified contours obtained by the DP algorithm, WM algorithm, and CCSM at the same scale.  
Figure 13: CCS-35 and CCS-2 under different mesh parameters. (**a**) CCS-35. (**b**) CCS-2.  
We obtained 100 simplified contours by intersecting 100 horizontal planes with CCS-35 and CCS-2 and then calculated the similarities between the simplified contours and C1. Figure 13(a) shows that when the target contour has 35 points, the trend of the similarity curve corresponding to each mesh parameter is roughly the same. Furthermore, when the mesh parameters were set to 0.001, 0.004, 0.006, and 0.008, these curves were almost similar and overlapping. When the mesh parameters were set to 0.002 and 0.01, some small local differences were observed among the curves, thereby suggesting that despite slight local differences, the curve trends of all mesh parameters were globally consistent. Figure 13(b) shows that when the target contour contains only the starting and ending points, the mesh parameters show similar and almost overlapping curve trends, which suggests that the similarity between the simplified and initial contours is insensitive to the mesh parameter, that is, mesh parameter does not affect the trend of similarity.  
Therefore, the similarity trend is generally consistent under different mesh parameters. The mesh parameters for CCSM have a relatively stable influence on the similarity assessment, especially for a wider scale range of multi-scale line simplifications. Moreover, the CCSM constructed by using a specific target contour has a certain influence on the similarity, but such influence can be alleviated by selecting appropriate mesh parameters. Given that a smaller mesh parameter corresponds to a higher computation cost, selecting a neutral value can balance the mesh size and quality of the solution.