## 4 Conclusions  
The Monte-Carlo stochastic method, spatial algebraic geometry theory and graphic visualization technique base on Open GL are used to model the three-dimensional DFN of the rock mass with rectangle as the fracture shape. The main discussion and conclusions are as follows:  
(1) A probabilistic model of geological parameters based on rectangular fractures was developed here using statistical theory, and the form of the coordinates of vertices of the rectangular fractures was derived;  
(2) The process of implementation of the simulation method of the rectangular DFN based on the Monte-Carlo technique was detailed using the MATALB platform with the Open GL graphics toolkit in the Visual C environment;  
(3) The simulation of DFN can only achieve \"probabilistic statistical simulation\", and it is difficult to achieve \"entity simulation\" at present. Therefore, the error in simulation of fracture distribution is inevitable, but it can be reduced by improving the accuracy of sampling, correcting sampling bias, and improving simulation algorithms;  
(4) Since the DFN is constructed based on the Monte-Carlo stochastic simulation method of probability theory, it is recommended to construct a certain number of DFNs that conform to the corresponding distribution laws in practice to study the fluctuation range of the mechanical properties of rock mass, and to provide a reliable basis for the correct evaluation of the quality of engineering rock mass.  
In summary, the results of the proposed model of the DFN were statistically similar to those obtained by on-site measurement in terms of the orientation, scale, and density of fractures in rock slope engineering on a representative 2D surface. This provides the foundation for subsequent research on the quantitative analysis of fractures in rock masses, such as the characteristics of their deformation and strength, the effects of the number of their dimensions, anisotropy, representative elementary volume, the identification of key blocks, the effects of excavation, and mechanisms of fracture of the rock mass. The DFN model in this manuscript is valid for close fractures, and open fractures will be considered in the next stage.  
Conceptualization, A.L. and G.S.; methodology, A.L. and F.W.; software, Y.L.; validation, Y.L., F.W. and Y.S.; formal analysis, A.L.; investigation, F.W.; resources, Y.L.; data curation, A.L.; writing--original draft preparation, Y.L.; writing--review and editing, A.L.; visualization, A.L.; supervision, G.S.; project administration, F.W.; funding acquisition, A.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (grant number 41672257); the Fundamental Research Funds for the Central Universities (grant number B200202051).  
Not applicable.  
The original contributions presented in the study are included in the article, and further inquiries can be directed to the corresponding authors.  
Thanks are due to [PERSON] of China railway No.17 bureau group sixth engineering Co., Ltd. for the section of application example.  
The authors declare no conflict of interest.