## 1 Introduction  
The grouping and orientation characterization of discontinuity (structural plane) sets is crucially important for rock mass characterization, geological disaster prevention and control, rock engineering applications (such as geothermal reservoirs, mines, tunnels, slopes, and water conservancy and hydropower), and earth science research [1, 2]. Over a long period of geological evolution, the crust has undergone complex geological processes of construction and transformation, and various types and scales of structural planes, such as joints, fractures, and faults, are widely developed in the natural rock mass [3, 4, 5, 6]. Because of the presence of discontinuity sets, the mechanical properties of rock masses exhibit significant anisotropic characteristics and are predominantly controlled by the structural plane. The occurrence, scale, density, morphology, and their combination of structural planes not only govern the stability of underground rock engineering but also dominate the shape, scale, and trend of the induced sliding body [7; 8], because essentially, the structural plane changes the stress distribution mode in the rock mass. Also, the accurate identification of structural planes contributes significantly to the classification of the rock mass and the subsequent prevention and control of engineering geological disasters. As a consequence, on-site measurement and statistics of structural plane parameters in rock masses to detect structural plane characteristics and their combination distribution patterns are necessary prerequisites for quantitative analysis and the calculation of rock mass engineering stability. In recent years, considerable efforts have been devoted to investigating effective methods for rock mass structural plane identification. Because of the strong randomness and uncertainty of the parameters of the structural plane, deterministic research methods are difficult to directly adopt. To analyze the regularity of developed structural planes, structural planes with some common characteristics are often classified, among which the most common is to group the structural planes and determine the dominant orientation according to their occurrence [9]. The traditional approaches for grouping structural planes generally adopt rose diagrams, pole plots, and isodensity plots. The representation of structural plane direction data and the identification of structural planes are usually conducted using techniques for the hemispherical projection of discontinuity poles (i.e., unit vectors with directions normal to structural planes) [10]. The advantage of this type of method lies in the ease of making an intuitive judgment on the distribution of the main structural planes, but the grouping results primarily depend on experience; in particular, the differences in results are even more pronounced when the boundaries between clusters are not clear [11; 12]. Moreover, the visual clustering method of calculating density contour lines by calculating the number of poles falling within the reference circle is a commonly used method, but due to sampling bias, this method has some problems that need to be corrected [13; 14]. In summary, these approaches may not be entirely satisfactory in some conditions, leading to the updating of alternative technologies for the automatic identification of structural planes. Numerous attempts to address subjective issues have already been made using cluster analyses. For instance, [PERSON] and [PERSON] [15] first proposed a clustering algorithm for structural plane orientation in 1976, which was then upgraded by [PERSON] and [PERSON] [16] and [PERSON] and [PERSON] [17] to develop an FCM clustering algorithm for the automatic identification of structural planes. [PERSON] et al. [18] described a random algorithm for clustering structural planes, which can handle the additional information provided; this algorithm is based on the probability distribution that defines each crack feature and involves the integration of these probability distribution functions.