## 2 Materials and Methods
### Identical Area Detection of Multi-Platform Point Cloud  
Although the TLS and HLS point clouds can be approximately transformed by the unified coordinate system, which provides good initial values for fine alignment, it is difficult to achieve correct convergence through the existing point cloud fine alignment algorithms due to the similarity of the features in an underground mine. For this reason, this paper firstly extracts the identical areas of the multi-platform point cloud, and then the alignment parameters of the identical areas are applied to the overall scanning scene, which realizes the unification of the coordinates of the multi-platform point cloud. In the field of mechanics, geometric elements such as the center of mass, the center of gravity, and curvature, and mechanical elements such as the moment of inertia are related to the spatial distribution of rigid body geometries. Physically, the moment of inertia corresponding to any rigid body around different axes is geometrically invariant. In other words, the axis characterizing the maximum or minimum moment of inertia of a rigid body is uniquely determined and will not change with the rotation or translation of the sampling points, which can be used as a geometric invariant to describe its attitude. As a result, the axis of the minimum moment of inertia can be used as the geometric reference to analyze the similarity of the distribution of the corresponding point cloud. The direction vector \(\mathbf{v}_{min}\) of the rotation axis corresponding to the minimum moment of inertia of the rigid body constituted by the discrete point cloud \(p_{i}=x_{i}\cdot e_{1}+y_{i}\cdot e_{2}+z_{i}\cdot e_{3}\) is:  
\[\mathbf{v}_{\min}=\text{argmin}[J(\mathbf{v})] \tag{5}\]  
where \(\|\mathbf{v}_{\min}\|=1\). In the 3D space coordinate system \(\text{O}-XYZ\), any axis \(\mathbf{v}\) through the origin O, whose unit direction vector is \(\mathbf{v}=x\cdot e_{1}+y\cdot e_{2}+z\cdot e_{3}\). The moment of inertia \(J(\mathbf{v})\) of a spatial rigid body consisting of discrete points around this axis \(\mathbf{v}\) is:  
\[J(\mathbf{v})=\sum_{i=1}^{u}m_{i}\Big{\|}(p_{i}\wedge\mathbf{v})\mathbf{v}^{-1}\Big{\|}^{2} \tag{6}\]  
where \(u\) is the number of point clouds and \(\wedge\) is known as the outer product in geometric algebra, which denotes the extension of one vector to another to obtain a dual vector or planar quantity. Assuming that the mass of the discrete point cloud \(p_{i}\) is 1, i.e., \(m_{i}=1\), considering the unit direction vector \(\mathbf{v}^{-1}=\mathbf{v}\) and \(\|\mathbf{v}\|=1\), the moment of inertia \(J(\mathbf{v})\) can be expressed as:  
\[J(\mathbf{v})=\sum_{i=1}^{u}\|(p_{i}\wedge\mathbf{v})\|^{2} \tag{7}\]  
where the two-paradigm number of the outer product of any discrete point cloud \(p_{i}\) and \(\mathbf{v}\) is:  
\[ \begin{array}{ll}\|p_{i}\wedge\mathbf{v}\|^{2}&=\|(x_{i}y-y_{i}x)e_{12}+(y_{i}z -z_{i}y)e_{23}+(z_{i}x-x_{i}z)e_{31}\|^{2}\\ &=(x_{i}y-y_{i}x)^{2}+(y_{i}z-z_{i}y)^{2}+(z_{i}x-x_{i}z)^{2}\end{array}  \tag{8}\]  
Then, the moment of inertia around the axis \(\mathbf{v}\) is:  
\[J(\mathbf{v})=\sum_{i=1}^{u}\Bigl{[}\left(x_{i}y-y_{i}x\right)^{2}+(y_{i}z-z_{i}y )^{2}+(z_{i}x-x_{i}z)^{2}\Bigr{]} \tag{9}\]  
Therefore, finding the minimum value of the moment of inertia is the problem of finding the extreme value of the objective Equation (5) subject to the constraints \(\mathbf{\varphi}(\mathbf{v})=\left\|\mathbf{v}\right\|^{2}-1=0\). Assuming the Lagrangian function \(L(\mathbf{v},\lambda)=J(\mathbf{v})+\lambda\mathbf{\varphi}(\mathbf{v})\), solve the partial derivatives concerning the variables \(x,y,z\) and \(\lambda\), which is shown in Equation (10):  
\[\left\{