## 5 Mirror Facet Alignment Method
### Three-Rotation Alignment Method  
In this section, a method to determine the angles of rotation \(\theta_{A}\), \(\theta_{B}\), and \(\theta_{C}\) is presented. When \(\mathbf{N}_{t}^{1}\) is parallel to the \(\mathbf{N}_{t}\), the angles of rotation \(\theta_{A}\), \(\theta_{B}\), and \(\theta_{C}\) can be calculated and facet alignment of the mirror unit can be achieved by rotation order CBA, such that the axis vector error is aligned. This is the three-rotation alignment method. The facet alignment process (Figure 5) includes the following steps:  
1. The mirror unit starts at facet 1 after initial installation and by adjusting bolt 3 is rotated according to the mirror unit angle \(\theta_{C}\) about axis \(\mathbf{e}_{\mathbf{A}\mathbf{B}}\), and point \(\mathbf{C}_{t}\) moves to point \(\mathbf{C}_{\text{AL}}\). 2. Bolt 2 is adjusted in order to rotate the mirror unit at angle \(\theta_{B}\) about axis \(\mathbf{e}_{\mathbf{C}_{\text{AL}}\mathbf{A}_{t}}\) and point \(\mathbf{B}_{t}\) is moved to point \(\mathbf{B}_{\text{AL}}\). 3. Bolt 1 is adjusted to rotate the mirror unit at angle \(\theta_{A}\) about axis \(\mathbf{e}_{\mathbf{B}_{\text{AL}}\mathbf{C}_{\text{AL}}}\) and point \(\mathbf{A}_{t}\) moves to point \(\mathbf{A}_{\text{AL}}\). After three adjustments, the vector \(\mathbf{N}_{t}\) becomes \(\mathbf{N}_{\text{AL}}\), \(\mathbf{Q}_{t}\) becomes \(\mathbf{Q}_{\text{AL}}\) and can be calculated by:  
\[\left\{ \begin{array}{l}\mathbf{Q}_{\text{AL}}=\mathbf{Q}_{t}\cdot\mathbf{H }_{\text{AL}}^{\text{CBA}}-\mathbf{A}_{t}\cdot\mathbf{H}_{\text{AL}}^{\text{ CBA}}+\mathbf{A}_{\text{AL}}\\ \mathbf{N}_{\text{AL}}=\mathbf{N}_{t}\cdot\mathbf{H}_{\text{AL}}^{\text{CBA}} \end{array} \right. \tag{17}\]  
Figure 5: The three-rotation alignment process of the mirror unit consists of: (**a**) rotation about axis \(\mathbf{A}_{t}\mathbf{B}_{t}\), (**b**) rotation about axis \(\mathbf{C}_{\text{AL}}\mathbf{A}_{t}\), and (**c**) rotation about axis \(\mathbf{B}_{\text{AL}}\mathbf{C}_{\text{AL}}\). where \(\mathbf{H}_{\mathrm{AL}}^{\mathrm{CBA}}=Rot(\mathbf{e}_{A_{\mathrm{t}}B_{\mathrm{t}}},\theta_{C})\cdot Rot(\mathbf{e}_{C_{\mathrm{AL}}A_{\mathrm{t}}},\theta_{B}) \cdot Rot(\mathbf{e}_{B_{\mathrm{AL}}C_{\mathrm{AL}}},\theta_{A})\ \ \mathbf{e}_{A_{ \mathrm{t}}B_{\mathrm{t}}}=(\mathbf{B}_{\mathrm{t}}-\mathbf{A}_{\mathrm{t}})/ \|\mathbf{B}_{\mathrm{t}}-\mathbf{A}_{\mathrm{t}}\|\); \(\mathbf{e}_{C_{\mathrm{AL}}A_{\mathrm{t}}}=(\mathbf{A}_{\mathrm{t}}-\mathbf{ C}_{\mathrm{AL}})/\|\mathbf{A}_{\mathrm{t}}-\mathbf{C}_{\mathrm{AL}}\|\); \(\mathbf{e}_{B_{\mathrm{AL}}C_{\mathrm{AL}}}=(\mathbf{C}_{\mathrm{AL}}- \mathbf{B}_{\mathrm{AL}})/\|\mathbf{C}_{\mathrm{AL}}-\mathbf{B}_{\mathrm{AL}}\|\); \(\mathbf{A}_{\mathrm{AL}}=(\mathbf{A}_{\mathrm{t}}-\mathbf{B}_{\mathrm{AL}}) \cdot Rot(\mathbf{e}_{B_{\mathrm{AL}}C_{\mathrm{AL}}},\theta_{A})+\mathbf{B} _{\mathrm{AL}}\); \(\mathbf{C}_{\mathrm{AL}}=(\mathbf{C}_{\mathrm{t}}-\mathbf{A}_{\mathrm{t}}) \cdot Rot(\mathbf{e}_{A_{\mathrm{t}}B_{\mathrm{t}}},\theta_{C})+\mathbf{A}_{ \mathrm{t}}\); \(\mathbf{B}_{\mathrm{AL}}=(\mathbf{B}_{\mathrm{t}}-\mathbf{A}_{\mathrm{t}}) \cdot Rot(\mathbf{e}_{C_{\mathrm{AL}}A_{\mathrm{t}}},\theta_{B})+\mathbf{A}_{ \mathrm{t}}\). The facet alignment error of the mirror unit can be described by error angle \(\theta\) and distance \(L\). The error angle \(\theta\) is the angle between the actual axis vector and the ideal axis vector \(\mathbf{N}_{0}\) and the error distance \(L\) is the distance between the actual and ideal vertex \(\mathbf{Q}\). After three rotations, \(\theta\) and \(L\) can be calculated by:  
\[\left\{ \begin{array}{c}L=\|\mathbf{Q}\vec{\mathrm{Q}}_{\mathrm{AL}}\|=\| \mathbf{Q}_{\mathrm{t}}\cdot\mathbf{H}_{\mathrm{AL}}^{\mathrm{CBA}}-\mathbf{A }_{\mathrm{t}}\cdot\mathbf{H}_{\mathrm{AL}}^{\mathrm{CBA}}+\mathbf{A}_{ \mathrm{AL}}\|\\ \theta=\arccos(\frac{\mathbf{N}_{\mathrm{AL}}\cdot\mathbf{N}_{0}}{\|\mathbf{ N}_{\mathrm{AL}}\|\cdot\|\mathbf{N}_{0}\|})\end{array} \right.