## 2 Principles of the RDFN Model  
Discrete fractures distributed in the natural rock mass are generally in complex shapes, as shown in Figure 0(a). Scholars have conducted simulations on complex joint network models for jointed rock masses with varying geometric shapes and spatial joint distributions to analyze their mechanical problems. The discrete fractures network (DFN) model is widely used as a joint characterization model to provide research ideas and characterization models for the mechanical properties of jointed rock mass [19]. This model has been widely used to simulate the mechanical properties of fractured rock mass, like the synthetic joint mass simulation. In this work, the DFN model is used to study the influence of structure planes on the failure pattern of the slope body.  
The probability statistics of the random distribution of discrete fracture networks underlie the DFN model. To simulate the model, a corresponding probability distribution function is established. Combining the joint density, joint trace length, and random seed of the simulated object can preliminarily generate the DFN model [20,21]. In the RDFN model proposed by [PERSON] et al. [18], its accompanying characterization methods are based on these three geometric types: curve, triangular broken line, and fractal fracture network model. Nonetheless, each RDFN model has its own geometric representation function. In order to standardize the model representation method, different geometric joint models are represented via various joint trace representation models based on the Fourier transform principle. Equation (1) selects the Fourier expansion of multi-series trapezoidal waves [18].  
\[f(x)=\frac{4A_{\text{max}}}{\pi wd}\sum_{n=1}^{\infty}\frac{\sin((2n-1)wd)}{ \left(2n-1\right)^{2}}\sin((2n-1)wx),n=1,2,3,. \tag{1}\]  
where \(T\) (as listed in Figure 0(b)) refers to the period length of the waveform, \(A_{\text{max}}\) refers to the amplitude height, and \(d\) refers to the rising width. The rising width determines the width of the trapezoidal platform which, in turn, further determines the geometric shape of the Fourier curve. The variable \(\omega\) determines the frequency of the curve, while \(n\) signifies the Fourier series of the curve.  
Figure 1: Principle of the Fourier-based RDFN model based on the complex joints in slope rock mass.  
Figure 1b displays the various geometric shapes of trapezoidal waves that correspond to Equation (1). Adjusting the aforementioned parameters enables obtaining a joint trace that conforms to the actual shape. Figure 1c shows the traditional slope model with smooth or linear joints, which neglects the roughness of the fractures. When considering the roughness of the joints, a more reliable slope model will be obtained, as shown in Figure 1d. Table 1 presents the Fourier parameters for diverse curves. Through modifying the d value of the Fourier trapezoidal wave, it is possible to establish a unified joint random discrete fracture network (RDFN) model for complex geometric shapes, such as straight lines, triangular broken lines, and rectangular steps. Using the MATLAB platform, Fourier joint models with varying trace lengths and certain densities are established based on information concerning linear discrete fracture network (DFN) joints. The primarying modeling methods are as follows: Firstly, generate Fourier curves with fixed trace lengths. Secondly, rotate them at specific angles and randomly translate them in certain areas to generate a set of linear joints. Thirdly, generate Fourier curves with different trace lengths and similarly rotate and translate them in the same area. Eventually, the RDFN joint models are generated. These joints cross each other and exhibit network characteristics.