## 3 Structural Plane Consistency Analysis between Boreholes  
In engineering practice, it is often necessary to conduct multiple borehole surveys when investigating the geology of an area. Since the structural plane has a special extension in space, the same structural plane can be penetrated by multiple boreholes through a reasonable arrangement of borehole locations. Collecting and analyzing the images of the inner wall of each borehole, and analyzing the consistency of the structural plane between boreholes by counting and filtering the structural plane data from the extracted images of multiple boreholes, is essential for engineering applications. When analyzing the structural plane consistency of multiple boreholes, it is assumed that the structural plane is straight and flat. Let the \(x\)-axis direct the due north direction, and a suitable space-rectangular coordinate system is established, as shown in Figure 3 for two boreholes A and B in space. The coordinates of their borehole centers are \((x_{1},y_{1},z_{1})\), and \((x_{2},y_{2},z_{2})\). Let vector \(\overrightarrow{m}\) is the unit vector of the axis of borehole A pointing in the positive direction of \(z\)-axis, and vector \(\overrightarrow{n}\) is the unit vector of the center of borehole B pointing in the positive direction of the \(z\)-axis. Let the angle between \(\overrightarrow{m}\) and the \(x\)-axis is \(\theta_{1}\) and the angle with \(z\)-axis is \(\theta_{2}\). \(\tau\)The angle between \(\overrightarrow{n}\) and the \(x\)-axis is \(\eta_{1}\), and the angle with the \(z\)-axis is \(\eta_{2}\)  
Figure 3: Consistency analysis diagram of the structural plane in boreholes. Figure 2: Structural plane in the expanding image of borehole wall. Then, \(\overrightarrow{m}=\{\cos\theta_{1}\sin\theta_{2},\ \sin\theta_{1}\sin\theta_{2},\ \cos \theta_{2}\}\), and \(\overrightarrow{n}=\{\cos\eta_{1}\sin\eta_{2},\ \sin\eta_{1}\sin\eta_{2},\ \cos \eta_{2}\}\). Therefore, we can write the straight-line equation of the axis of the borehole A as  
\[\frac{x-x_{1}}{\cos\theta_{1}\sin\theta_{2}}=\frac{y-y_{1}}{\sin\theta_{1} \sin\theta_{2}}=\frac{z-z_{1}}{\cos\theta_{2}} \tag{2}\]  
As shown in Figure 4,the structural plane in borehole A is \(s_{i}\). Its inclination is measured to be \(\alpha_{i}\), and the dip angle is \(\beta_{i}\). The depth of the borehole is hi (the depth of the borehole is the distance from the structural plane to the hole opening, the same below), and the surface normal vector of the structural plane is \(\overrightarrow{n_{i}}\). We can obtain \(\overrightarrow{n_{i}}=\ \{\sin\alpha_{i}\sin\beta_{i},\cos\alpha_{i}\sin\beta_{i}, \cos\beta_{i}\}\), where \(\beta_{i}\
eq\pi/2\). Correspondingly, one of the structural planes in borehole B is \(s_{j}\), its inclination is measured to be \(\alpha_{j}\) and the dip angle is \(\beta_{j}\), and the depth of the borehole is \(h_{j}\). Set the origin to O, OP is perpendicular to the plane of structural plane \(s_{i}\), and the vertical foot is point P. The vertical line OQ is perpendicular to the surface of structure plane \(s_{j}\), and the perpendicular point is Q, since the depth of structural plane \(s_{i}\) in the borehole is \(h_{i}\). The intersection of the axis of borehole A and the structural plane is M. The coordinates of M are \((x_{M},y_{M},z_{M})\), where the value of \(x_{M},y_{M}\) and \(z_{M}\) are Equations (3)-(5). \[x_{M}=h_{i}\cos\theta_{1}\sin\theta_{2}+x_{1} \tag{3}\]  
\[y_{M}=h_{i}\sin\theta_{1}\sin\theta_{2}+y_  
small, some corresponding surfaces may be missed during the screening. Therefore, when verifying the consistency of the structure planes, it is crucial to determine a suitable \(\varepsilon\) is crucial. After the above calculation, we can initially judge the consistency of the structural plane \(s_{i}\) in Borehole A and the structural plane \(s_{j}\) in Borehole B.