# Photogrammetry-Based 3D Textured Point Cloud Models Building and Rock Structure Estimation
## 5 The Rock Structure Analysis for the Rock Slope
Second, joint density is calculated according to the relative position of joints and sampling windows on the outcrop. Third, the diameters of joint disks are inferred based on the distribution of the trace lengths. Fourth, the orientations of joint disks are generated on the basis of the joint density and the orientation distribution of joints on the outcrop. It should be noted that all the steps are executed based on the data of joints collected on the outcrop, and steps 2 to 4 are carried out by the joint group. A total of 2993 traces on the rock slope were artificially extracted from the textured 3D point cloud model in Meshlab. The study involved a manual process of extracting trace lines by identifying multiple inflection points along a single trace line, connecting them, and obtaining the final trace line. The software currently cannot automatically discriminate between lines and areas in 3D point cloud models, and this work was done manually instead. It is preferred that joints be automatically extracted and identified in the future. The lengths of the traces range from 0.01 to 12.16 m. Figure 7 shows the distribution of the traces on the slope. Each of the traces consists of dozens of inflection points, the coordinates of which are the basis of the joint parameter calculation. The orientation (including the dip and dip direction) of a joint can be inferred based on at least three non-collinear points on the joint. Joints were grouped according to their orientations. The orientation is composed of two parameters: dip and dip direction. Fuzzy C-means was used to execute the grouping when the number of joint sets ranged from 2 to 8 ([PERSON] et al., 2016). The number of joint sets was determined according to a simplified Xie-Beni index, shown as Equation (17) ([PERSON] et al., 2017). The index  
Figure 7: Trace lines extracted from the textured 3D point cloud model. has the advantages of simple programming, high precision, and good practicability [26]. The number of joint sets that minimize the index is the optimum number of joint sets. \[SXB_{K}=\frac{\sum\limits_{i=1}^{N}\sum\limits_{vj=1}^{K}I_{ivj}\left(\arccos \left|x_{i}x_{vj}+y_{i}y_{vj}+z_{i}z_{vj}\right|\right)^{2}}{\min\limits_{vj \
eq vl}\left(\arccos\left|x_{vj}x_{vl}+y_{vj}y_{vl}+z_{vj}z_{vl}\right|\right)^ {2}} \tag{17}\]  
where \(SXB_{K}\) is the simplified Xie-Beni set validity index when the \(N\) joints are divided into \(K\) sets. (\(x_{i}\), \(y_{i}\), \(z_{i}\)) are coordinates of the unit normal vector of joint \(i\). (\(x_{vj}\), \(y_{vj}\), \(z_{vj}\)) and (\(x_{vl}\), \(y_{vl}\), \(z_{vl}\)) are coordinates of the unit normal central vectors of sets \(vj\) and \(vl\), respectively. \(I_{ivj}\) is a Boolean value, which is equal to 1 if the joint \(i\) belongs to the set \(vj\); otherwise, it is equal to 0. Joints were grouped into three sets. Figure 8 is the poles diagram of the grouping results for the 2993 traces. One pole represents the orientation of one joint, and the angle between the pole and North represents the dip direction, while the distance between the pole and the circle center represents the magnitude of the dip. Because the surface of a rock mass is opaque, the joint density and diameters of joint disks were inferred using 3D topographic sampling windows, which allow consideration of joint traces within buffer zones [18]. Due to the fact that extracted traces in the 3D point cloud model are three-dimensional curves, and the actual slope surface is also a three-dimensional surface, the buffer zone method is adopted in this paper for trace information statistics. This method converts the linear boundary of the circular measuring window into a surface boundary, counts the trace information within and intersecting with the buff zone boundary, and then calculates the distribution parameters of the joint diameter and density. Finally, a joint disk model can be generated based on these parameters. Figure 9 shows the traces of joints by the set and the employed 3D topographic sampling windows.