## 3 Measurement of the Mirror Facet Alignment
### Mathematical Model of the Mirror Facet
\begin{array}{ll}\overset{\rightarrow}{\mathrm{T}_{1t }\mathrm{T}_{3}^{1}\times\overset{\rightarrow}{\mathrm{T}_{1t}\mathrm{T}_{3t }}}&\mathbf{T}_{3}^{1}\
eq\mathbf{T}_{3t}\\ \overset{\rightarrow}{\mathrm{\left\|T_{1t}\mathrm{T}_{3}^{1}\times\mathrm{T}_ {1t}\mathrm{T}_{3t}\right\|}}&\mathbf{T}_{3}^{1}\
eq\mathbf{T}_{3t}\\ \left[1,0,0\right]&\mathbf{T}_{3}^{1}=\mathbf{T}_{3t}\end{array} \right. \tag{6}\]  
According to the motion parameters (\(\mathbf{n}_{1}\) and rotation angle \(\varphi_{3}\)) of the mirror unit, \(\mathbf{p}_{2}\) and \(\mathbf{N}_{1}\) can be deduced as:  
\[\left\{ \begin{array}{ll}\mathbf{p}_{2}=(\mathbf{p}_{1}-\mathbf{T}_{1t})Rot( \mathbf{n}_{1},\varphi_{3})+\mathbf{T}_{1t}\\ \mathbf{N}_{1}=\mathbf{N}\cdot Rot(\mathbf{n}_{1},\varphi_{3})\end{array} \right. \tag{7}\]  
Step 3: The mirror unit rotating about the axis \(\mathrm{T}_{1t}\mathrm{T}_{3t}\) (unit vector of rotation axis \(\mathbf{n}_{2}=\mathrm{T}_{1t}\mathrm{T}_{3t}/\|\mathrm{T}_{1t}\mathrm{T}_{3t }\|\)) causes point \(\mathrm{T}_{2}^{2}\) to rotate to position \(\mathrm{T}_{2t}\) (Figure 3c). At this time, point \(\mathrm{p}_{2}\) moves to \(\mathrm{p}_{3}\) and \(\mathbf{N}_{1}\) becomes \(\mathbf{N}_{3}\). The mirror unit thus has moved from the ideal facet to facet 1. In Figure 3c, the distance between axis \(\mathrm{T}_{1t}\mathrm{T}_{3t}\) and point \(\mathrm{T}_{2}^{2}\) is \(\|\mathrm{HT}_{2}^{2}\|\), which satisfies the \(\overset{\rightarrow}{\mathrm{\left\|HT}_{2}^{2}\right\|}=\|\mathrm{HT}_{2 }^{\rightarrow}\|\). Again, according to the triangle cosine theorem, the rotation angle \(\varphi_{2}\) can be deduced as:  
\[\varphi_{2}=\mathrm{arccos}\left(\frac{\overset{\rightarrow}{2}\overset{ \rightarrow}{\mathrm{\left\|HT}_{2}^{2}\right\|}-\|\Gamma_{2t}^{2}\mathrm{T}_ {2t}\|}{\overset{\rightarrow}{\mathrm{\left\|HT}_{2}^{2}\right\|}}\right) \tag{8}\]  
According to the motion parameters of mirror unit in **step 3**, the vectors \(\mathbf{p}_{3}\) and \(\mathbf{N}_{3}\) can be deduced as:  
\[\left\{ \begin{array}{ll}\mathbf{p}_{3}=(\mathbf{p}_{2}-\mathbf{T}_{1t})Rot( \mathbf{n}_{2},\varphi_{2})+\mathbf{T}_{1t}\\ \mathbf{N}_{3}=\mathbf{N}\cdot Rot(\mathbf{n}_{1},\varphi_{3})\cdot Rot( \mathbf{n}_{2},\varphi_{2})\end{array} \right. \tag{9}\]  
where \(\mathrm{T}_{2}^{\rightarrow}\mathrm{T}_{2t}=\mathbf{T}_{2t}-\mathbf{T}_{2}^{2}\), \(\mathbf{T}_{2}^{2}=(\mathbf{T}_{2}+\mathbf{t}_{1}-\mathbf{T}_{1t})Rot( \mathbf{n}_{1},\varphi_{3})+\mathbf{T}_{1t}\). Putting Equation (4) and Equation (7) into Equation (9), the vector \(\mathbf{p}_{3}\) can be rewritten as:  
\[\mathbf{p}_{3}=\left(\mathbf{p}_{0}-\mathbf{T}_{1}\right)\cdot Rot(\mathbf{n} _{1},\varphi_{3})\cdot Rot(\mathbf{n}_{2},\varphi_{2})+\mathbf{T}_{1}+ \mathbf{t}_{1} \tag{10}\]  
where \(\mathbf{N}\) in Equation (9) and \(\mathbf{p}_{0}\) in Equation (10) are both arbitrary. Putting the vertex position of vector \(\mathbf{Q}\) and \(\mathbf{N}_{0}\) to the Equation (9) and Equation (10) respectively, the vertex position vector \(\mathbf{Q}_{t}\) and axis vector \(\mathbf{N}_{t}\) can be determined as:  
\[\left\{ \begin{array}{ll}\mathbf{Q}_{t}=(\mathbf{Q}-\mathbf{T}_{1})\cdot Rot (\mathbf{n}_{1},\varphi_{3})\cdot Rot(\mathbf{n}_{2},\varphi_{2})+\mathbf{T}_ {1}+\mathbf{t}_{1}\\ \mathbf{N}_{t}=\mathbf{N}_{0}\cdot Rot(\mathbf{n}_{1},\varphi_{3})\cdot Rot( \mathbf{n}_{2},\varphi_{2})\end{array} \right. \tag{11}\]  
Similarly, by replacing \(\mathbf{p}_{0}\) in Equation (10) with vector \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\), the position vector \(\mathbf{A}_{t}\), \(\mathbf{B}_{t}\) and \(\mathbf{C}_{t}\) can be determined as:  
\[\left\{