# A Global Continuous Multi-Scale Terrain Contour Simplification Method Based on the Level Set
## 2 Methodology
### Generating Global Continuous Multi-Scale Simplified Contours  
#### 2.3.1 Generation of Simplified Contours  
Contour lines are essential parts of a terrain model. In terrain modeling, multiple levels of contours can be modeled on the 3D surface, and each contour depicts a terraced representation of the 3D surface at an elevation level. To obtain simplified contours, the CCS was intersected with a set of horizontal planes with successive height values to generate intersection lines. These intersection lines were taken as global continuous multi-scale simplified contours with varying simplification degrees of the initial contour.  
To calculate the intersection lines between CCS and a set of horizontal planes, the CCS was tessellated into a polygonal mesh (e.g., triangle or quadrangle), that is, the \(\left[u_{i},v_{j}\right]\), values of NURBS were mapped to the corresponding grid points of the polygonal mesh. The triangle mesh was used to generate a polygonal mesh due to its advantages in the rapid generation and its intersection with planes. As shown in Figure 5, the triangle mesh comprises a set of triangular faces covering CCS and was therefore labeled CCS-mesh. This mesh may become irregular due to the uneven distribution of unconstrained sample grid points. To ensure the stability and convergence of the numerical solution, a suitable mesh generation technique should be applied to construct the appropriate surface mesh and to maintain a well-defined mesh that fits well in the CCS throughout the intersection process.  
Based on the CCS-mesh, we built a set of equidistant horizontal planes to intersect the CCS-mesh and the intersection lines that are regarded as simplified intermediate results. As defined in Equation (6) below, the result of every horizontal plane intersecting the CCS-mesh is an intersection line \(L_{(Z_{i})}\). The height \(Z_{i}\) corresponds to each horizontal plane and is between the elevations of \(Z_{min}\) and \(Z_{max}\). An intersection contour cluster was obtained by iterating the intersection between CCS-mesh and every horizontal plane. These intersecting  
Figure 4: Established CCS.  
Figure 5: CCS-mesh covered by triangular faces.  
lines were considered simplified contour lines. The scale corresponding to these simplified contour lines continuously varied between the scales of the initial and target contours.  
\[\begin{array}{c}\text{Contour cluster }=\left\{L_{(Zi)}\mid L_{(Zi)}=CCS_{\text{mesh}} \ \cap\ \ \text{Cutting plane }(Z_{i})\right.\\ \left.Z_{i}=Z_{min}+i*\ \text{interval, }i\leq\text{round}(Z_{max}-Z_{min})/ \text{interval}\right\}\end{array} \tag{6}\]  
Figure 6 presents an example of the simplified intermediate results generated by the CCS-mesh intersecting with a set of horizontal planes. Figure 6a represents a set of equidistant horizontal planes with continuous elevation; Figure 6b shows the intersection between the CCS-mesh and planes; Figure 6c illustrates the global continuous multiscale transformation of intersection lines. Each intersection line between the CCS-mesh and a horizontal plane captured the morphology characteristics of contours at a certain height level. Therefore, this cluster of simplified contours preserves the morphology characteristics corresponding to the continuously changing scales between the initial and target contours. The location or number of horizontal planes can be determined based on the desired scale simplified results or the number of simplified contours demanded by users.  
Figure 6: Intersection between the CCS-mesh and horizontal planes. (**a**) A set of horizontal planes. (**b**) The intersection between CCS-mesh and horizontal planes. (**c**) The simplified contours on horizontal planes.