## 2 Rectangular Pipe Jacking Excavation Cross-Section Modeling
### Conformal Transformation  
Conformal transformation is the process of transforming a region with complex boundary shapes into a region with simple boundary shapes. The result is to transform an arbitrary region in the \(z\)-plane into a simple region in the \(\zeta\)-plane, such as within or outside a unit circle. This method is particularly effective for solving the problems of holes. As shown in Figure 3, the approximate mapping function using the series form [28] is:  
\[z=\omega(\zeta)=\mathrm{i}\lambda\frac{1+\zeta}{1-\zeta}+\mathrm{i}\sum_{k=1} ^{n}\beta_{k}(\zeta^{k}-\frac{1}{\zeta^{k}}) \tag{1}\]  
The \(z=x+\mathrm{i}y\) plane is mapped to the \(\zeta=\zeta+\mathrm{i}\eta\) plane, where \(\mathrm{i}\) is an imaginary unit, \(\beta_{k}\) (\(k=1,2,3,\ldots,n\)) are real constants to be determined, and \(n\) is the number of terms in the series. In the \(\zeta\)-plane, any point \(\zeta\) can be expressed as polar coordinates (\(\rho,\theta\)). It can be seen that the ground surface is mapped as an (outer) circular annulus of radius 1, the excavation boundary is mapped as an (inner) circular annulus of radius \(\alpha<1\), and the points A, B, C, F, D, and E are mapped as points A\({}^{\prime}\), B\({}^{\prime}\), C\({}^{\prime}\), F\({}^{\prime}\), D\({}^{\prime}\), and E\({}^{\prime}\), respectively.  
Using polar coordinates (\(\rho\), \(\theta\)) in the \(\zeta\)-plane, it is known that \(\zeta=\rho\mathrm{e}^{\mathrm{i}\theta}\). In order to determine the parameters \(\lambda\) and \(\beta_{k}\) (\(k=1,\,2,\,3,\ldots,\,n\)) in the conformal transformation Equation (1), for the \(z\)-plane, it is known that \(z=x+\mathrm{i}y\). For the pipe jacking excavation boundary, starting from point C to the end of point F, these two points are the necessary control points, which determine the headroom and the buried depth of the tunnel, and a total of \(m\geq 2\) control points need to be selected. The boundary corresponds to \(\rho=\alpha\) (\(\alpha<1\)) in the \(\zeta\)-plane, and it is known that \(\zeta=\alpha\sigma=\alpha\mathrm{e}^{\mathrm{i}\theta}\). The \(x\) and \(y\) coordinates of each control point can be expressed using Equation (1), respectively, as:  
\[x_{j}=-\frac{2\lambda\alpha\sin\theta_{j}}{1+\alpha^{2}-2\alpha\cos\theta_{j} }-\sum_{k=1}^{n}\beta_{k}(\alpha^{k}+\frac{1}{\alpha^{k}})\sin(k\theta_{j}) \tag{2}\]  
\[y_{j}=\frac{\lambda(1-\alpha^{2})}{1+\alpha^{2}-2\alpha\cos\theta_{j}}+\sum_ {k=1}^{n}\beta_{k}(\alpha^{k}-\frac{1}{\alpha^{k}})\cos(k\theta_{j}) \tag{3}\]  
where \(j\) (\(j=1,\,2,\,3,\ldots,\,m\)) is the number of the control point.  
Since points C and F are the necessary control points, which correspond to \(\theta_{1}=0^{\circ}\) and \(\theta_{m}=180^{\circ}\) of the \(\zeta\)-plane, by combining Equations (2) and (3), \(2m-2\) independent equations  
Figure 3: Plane of conformal transformation.  
can be obtained for the \(m\) control points. There are \(n+m\) independent parameters in total including \(\lambda\), \(\alpha\), \(\beta_{k}\) (\(k=1\), \(2\), \(3\),..., \(n\)), and \(\theta_{j}\) (\(j=2\), \(3\),..., \(m-1\)), respectively. Therefore, taking \(m=n+2\), the number of independent equations is equal to the number of independent parameters. Solving this system of independent equations yields all the independent parameters, thus determining the mapping function shown in Equation (1).