## 2 Deformation Monitoring of Inverted Arch Uplifts of Soft Rock Tunnels
### Real-Time or Regular Deformation Data Solution  
According to the three-dimensional coordinates of all points in the inverted arch slice point cloud of soft rock tunnels, a simple mathematical calculation is carried out, and the abscissa, ordinate, and vertical coordinates are averaged to obtain the center of gravity ofthe point cloud slice. By extending the idea of the point to line distance formula to calculate the distance from the point to the plane, we can obtain:  
\[D_{i^{\prime}}=\frac{\left|\hat{a}x_{i^{\prime}}\ +\ \hat{b}y_{i^{\prime}}\ +\ \hat{c}z_{i^{\prime}}\right|}{\sqrt{\left(\hat{a}\ +\ \hat{b}\ +\ \hat{c}\right)^{2}}} \tag{24}\]  
The distance can be obtained by using Formula (24), that is, the distance between the point at any end of the point cloud and the datum plane. At this time, the distance between the first slice point cloud to be intercepted and the datum plane is set, and then the 3D coordinate data of the initial slice point cloud can be obtained and are recorded as \(\left(x_{i^{\prime}}^{1},y_{i^{\prime}}^{1},z_{i^{\prime}}^{1}\right)\). The barycentric coordinates of the initial slice point cloud can be expressed as:  
\[\left\{\begin{array}{l}\overline{x}=\frac{\sum\limits_{i^{\prime}=1}^{ \varnothing}x_{i^{\prime}}^{1}}{\omega D_{i^{\prime}}}\\ \overline{y}=\frac{\sum\limits_{i^{\prime}=1}^{\varnothing}y_{i^{\prime}}^{1 }}{\omega D_{i^{\prime}}}\\ \overline{z}=\frac{\sum\limits_{i^{\prime}=1}^{\varnothing}z_{i^{\prime}}^{1 }}{\omega D_{i^{\prime}}}\end{array}\right. \tag{25}\]  
where \(\varnothing\) is the total number of points included in the point cloud of the inverted arch slice of the initial soft rock tunnel.  
The 3D coordinate data and barycentric coordinate calculation method of the initial point cloud slice are extended to the \(\mu\) point cloud of an inverted arch slice of soft rock tunnel \(\mu\). The 3D coordinate data of point cloud slices can be marked as \(\left(x_{i^{\prime}}^{\mu},y_{i^{\prime}}^{\mu},z_{i^{\prime}}^{\mu}\right)\). The barycentric coordinates are \(\left(\overline{x}_{i^{\prime}},\overline{y}_{i^{\prime}},z_{i^{\prime}}\right)\). The calculation of the distance between the center of gravity and the uplifted ground of the inverted arch in the soft rock tunnel, that is, the distance between the bottom and the ground, will become a simple mathematical problem. According to the point to plane and the distance calculation formula, the shape, curvature, deformation and other parameters of the inverted arch will be solved one by one.  
In the ground fitting formula for reverse arch uplift in soft rock tunnels, the ground plane parameters are calibrated, and can replace the ground shape of the reverse arch in soft rock tunnels. By applying these parameters to the calculation process, real-time or regular deformation data of reverse arches in soft rock tunnels can be obtained, and the internal structure and deformation situation of soft rock tunnels can be more intuitively understood. By using ground plane parameters, the shape of reverse arches in soft rock tunnels can be inferred. Parameters such as the curvature and deformation are used for deformation monitoring.