## 2 Methods
### Calculation of Centroid Points of Each Block and Side Points  
Because soil at the cliff and ground objects on the soil require bearing capacity support, that is, the bottom of the cliff near the sea side is mostly hollowed out and there is less ground object support in the vertical direction, the possible location of the support under the cliff must be determined. According to the analysis of terrain characteristics and the supporting effect of force on the cliff, supports under the cliff can be offset from the vertical direction under the cliff to the land side. According to the stress characteristics of ground objects, the average mass of the objects should be concentrated at its center of mass; that is, for the cliff, the section from the vertical direction under the cliff to the center of mass of the cliff can fully support the cliff. If there is no support in this section, this section is hollow, the cliff cannot be supported, and it will fall. Therefore, the direction of the center of mass under the cliff is the limiting direction for the stress support of the cliff. However, in the section from the center of mass to the water, the support under the cliff may have many different ground object shapes. Therefore, the coastline under the cliff may have various shapes. In this case, measuring the coastline under the cliff is difficult using a single standard. Therefore, to complete the coastline, its part under the cliff should not be missing. The centroid direction of the cliff can be regarded as an estimated direction, and as a better estimate of coastline under the cliff, coastline obtained in this direction can be  
Figure 4: The schematic diagram of point cloud segmentation, centroid points, and side points.  
used as the innermost coastline under the cliff. Center typically refers to the center of mass. Presumably, all masses of an object are concentrated at this point. The coordinates of the center \(O_{c}\) of an object can be calculated using the following formula:  
\[O_{c}=\frac{1}{M}\sum_{n}^{i=0}m_{i}r_{i} \tag{3}\]  
where, \(r_{i}=(x_{i},y_{i},z_{i}),\ i=1,2,\cdots,n\) is the coordinate of each particle, and \(m_{i}\) is the corresponding mass of the particle.  
The point cloud library (PCL) platform and the developed procedure are used to calculate the centroid of the point cloud. When calculating the centroid of the point cloud, \(m_{i}=1\) is required in Equation (3). Then, the centroid coordinates of the point cloud can be calculated as follows:  
\[O_{c}=\frac{1}{n}(\sum_{i=0}^{n}x_{i},\sum_{i=0}^{n}y_{i},\sum_{i=0}^{n}z_{i}) \tag{4}\]  
As shown in Figure 4, the centroid coordinates of the point clouds of each block can be calculated using Equation (4). If each block assumes the point projected onto the water surface by the centroid point as the representative point, the number of points is small; thus, accurately representing the situation is challenging in this case. Therefore, if the vertical line of the two boundaries of each block passing through each centroid point intersects the two boundaries, the x coordinate of the two intersections is the x coordinate of the two boundaries, their y coordinate is the y coordinate of the centroid point, and their z coordinate is the z coordinate of the centroid. In this manner, each block point cluster was built with two side points, all of which were stored in the point cloud dataset and labeled Pside. This method has no specific requirements on the density of the point cloud, and has good applicability to all kinds of point cloud data. However, the higher the density of the point cloud, the higher the subsequent coastline extraction accuracy will be; otherwise, the coastline accuracy will be reduced.