## 3 Measurement of the Mirror Facet Alignment
### Mathematical Model of the Mirror Facet  
As previously stated, to calculate the facet parameters of the mirror unit, the OneT-TwoR method was developed (Figure 3). Briefly the method determines facet 1 based on the position of the mirror unit as it arrives at after one translational motion and two rotational motions. Therefore, the position vectors \(\mathbf{T}_{1}\), \(\mathbf{T}_{2}\), \(\mathbf{T}_{3}\) and \(\mathbf{T}_{1t}\), \(\mathbf{T}_{2t}\), \(\mathbf{T}_{3t}\) before and after installation of the mirror unit respectively, which could be used to determine the motion parameters of the OneT-TwoR method. According to motion parameters of the OneT-TwoR method, we can then calculate \(\mathbf{Q}_{t}\) and \(\mathbf{N}_{t}\). The OneT-TwoR process of mirror unit and the deduction of mirror facet mathematical model could follow the Step 1, Step 2, and Step 3 in this section. In Figure 3a, the point \(\mathrm{p}_{0}\) (position vector \(\mathbf{p}_{0}\)) and axis vector \(\mathbf{N}\) are located in the ideal position to describe rigid body motion and derive the formulas. To derive the formula, a function \(Rot(\mathbf{e},\beta)\) for determining vector \(\mathbf{P}\in\mathbf{R}^{1\times 3}\) rotating around the unit vector \(\mathbf{e}=[e_{x},e_{y},e_{z}]\) with \(\beta\) degrees to get the vector \(\mathbf{P}^{\prime}\) can be defined by [31]:  
\[\mathbf{P}^{\prime}=\mathbf{P}\cdot\cos\beta+(\mathbf{e}\cdot\mathbf{P}^{ \mathrm{T}})\cdot\mathbf{e}(1-\cos\beta)+(\mathbf{e}\times\mathbf{P})\sin \beta=\mathbf{P}\cdot Rot(\mathbf{e},\beta) \tag{2}\]  
where superscript \(\mathrm{T}\) denotes the transpose. Furthermore, by expanding and reducing the middle section of the Equation (2), the function \(Rot(\mathbf{e},\beta)\) can be given by:  
\[Rot(\mathbf{e},\beta)=\left[ \begin{array}{ccc}\cos\beta+e_{x}^{2}(1-\cos \beta)&e_{x}e_{y}(1-\cos\beta)+e_{z}\cdot\sin\beta&e_{x}e_{z}(1-\cos\beta)-e_{ y}\cdot\sin\beta\\ e_{x}e_{y}(1-\cos\beta)-e_{z}\cdot\sin\beta&\cos\beta+e_{y}^{2}(1-\cos\beta)&e_ {y}e_{z}(1-\cos\beta)+e_{x}\cdot\sin\beta\\ e_{x}e_{z}(1-\cos\beta)+e_{y}\cdot\sin\beta&e_{y}e_{z}(1-\cos\beta)-e_{x}\cdot \sin\beta&\cos\beta+e_{z}^{2}(1-\cos\beta)\end{array} \right] \tag{3}\]  
Step 1: The translational motion of the mirror unit causes point \(\mathrm{T}_{1t}\) move to point \(\mathrm{T}_{1t}\) (Figure 3a) and at the same time, \(\mathrm{p}_{0}\) moves to \(\mathrm{p}_{1}\). The position vector \(\mathbf{p}_{1}\) can then be given as:  
\[\mathbf{p}_{1}=\mathbf{p}_{0}+\mathbf{t}_{1} \tag{4}\]  
Step 2: The mirror unit rotating about point \(\mathrm{T}_{1t}\) (unit vector of rotation axis is \(\mathbf{n}_{1}\)) causes point \(\mathrm{T}_{3}^{1}\) to rotate to point \(\mathrm{T}_{3t}\). At this time, point \(\mathrm{p}_{1}\) moves to \(\mathrm{p}_{2}\) and \(\mathbf{N}\) becomes \(\mathbf{N}_{1}\). The rigid body motion satisfied the rules \(||\mathrm{T}_{1t}\mathrm{T}_{3}^{1}||=||\mathrm{T}_{1t}\mathrm{T}_{3t}||=|| \mathrm{T}_{1}\mathrm{T}_{3}||\), where \(||\mathbf{A}||\) is magnitude of the vector \(\mathbf{A}\). According to the triangle cosine theorem, the rotation angle \(\varphi_{3}\) can be deduced as:  
\[\varphi_{3}=\arccos\left(\frac{2||\mathrm{T}_{3}-\mathrm{T}_{1}||^{2}-|| \mathrm{T}_{3t}-\mathrm{T}_{3}^{1}||^{2}}{2||\mathrm{T}_{3}-\mathrm{T}_{1}||^{ 2}}\right) \tag{5}\]  
Figure 3: The mirror unit through OneT-TwoR rigid body motion as it moves from the ideal facet to facet 1 by (**a**) translational motion, (**b**) rotation around point \(\mathrm{T}_{1t}\), and (**c**) rotation about axis \(\mathrm{T}_{1t}\)\(\mathrm{T}_{3t}\). The unit vector \(\mathbf{n}_{1}\) is perpendicular to both vector \(\mathrm{T}_{1t}\mathrm{T}_{3}^{1}\) and vector \(\mathrm{T}_{1t}\mathrm{T}_{3t}\) and can be calculated by:  
\[\mathbf{n}_{1}=\left\{