## 4 Equivalent of the Facet 1  
The movement process of a rigid body is reversible, i.e., a rigid body that can move from facet 1 to facet 2 can also take the reverse path from facet 2 to 1. In this paper, the facet alignment process of a mirror unit is taken as a three-rotation movement and the axis of rotation consists of two ball-hinge center points (Section 2). The OneT-TwoR rotational movement does not correspond to the alignment process of the mirror facet, as in Section 3.2, because \(\mathbf{n}_{1}\) may not consist of two ball-hinge center points, as shown in step 2. Thus, a three rotation one translational rigid body movement (ThreeR-OneT) to get to facet 1 (Figure 4) was developed according to the facet alignment process of a mirror unit with three-rotational movement. This means facet 1 of the mirror unit arrives at this position from the ideal position through three rotational motions and one translational motion (Figure 4). As such, the facet alignment error can be reduced or eliminated by reversing the order of motion. Theoretically, three rotational motions of mirror unit can occur in six different orders: ABC (rotate point A, B, and C in turn, see Figure 4), ACB, BAC, BCA, CAB, or CBA. In this section, taking the order ABC as an example the mathematical model of the mirror facet after the ThreeR-OneT motion was established. The process for the ThreeR-OneT motion of the mirror unit is as follows:  
1. The mirror unit located at the ideal facet, then rotate the mirror unit angle \(\theta_{A}\) around the axis of CB (Figure 4a). 2. The mirror unit is rotated at angle \(\theta_{B}\) around axis A\({}_{\text{t}}^{1}\)C(Figure 4b). 3. The mirror unit is rotated at angle \(\theta_{C}\) around axis B\({}_{\text{t}}^{1}\)A\({}_{\text{t}}^{1}\)(Figure 4c). 4. The mirror unit is translated by the vector \(\mathbf{M}_{1}=[x,y,z]\). Moreover, additional mathematical models of rotational order can be established by following the above steps. According to the mathematical model derived in Section 3.2, the position vector of point P (Figure 4a) after each movement phase can be derived as:  
\[\left\{ \begin{array}{l}\mathbf{P}_{1}=(\mathbf{P}-\mathbf{C})\cdot Rot( \mathbf{e_{CB}},\theta_{A})+\mathbf{C}\\ \mathbf{P}_{2}=(\mathbf{P}_{1}-\mathbf{C})\cdot Rot(\mathbf{e_{A_{t}^{1}C^{ \prime}}},\theta_{B})+\mathbf{C}\\ \mathbf{P}_{3}=(\mathbf{P}_{2}-\mathbf{B}_{\text{t}}^{1})\cdot Rot(\mathbf{e_{ B_{t}^{1}A_{t}^{1}}},\theta_{C})+\mathbf{B_{t}^{1}}\\ \mathbf{P}_{4}=\mathbf{P}_{3}+\mathbf{M}_{1}\end{array} \right. \tag{13}\]  
where \(\mathbf{e_{CB}}\), \(\mathbf{e_{A_{t}^{1}C^{\prime}}}\) and \(\mathbf{e_{B_{t}^{1}A_{t}^{1}}}\) are the unit vectors of the rotational axis and can be defined as: \(\mathbf{e_{CB}}=(\mathbf{B}-\mathbf{C})/\|\mathbf{B}-\mathbf{C}\|\), \(\mathbf{e_{A_{t}^{1}C}}=(\mathbf{C}-\mathbf{A_{t}^{1}})/\|\mathbf{C}-\mathbf{A _{t}^{1}}\|\), \(\mathbf{e_{B_{t}^{1}A_{t}^{1}}}=(\mathbf{A_{t}^{1}}-\mathbf{B_{t}^{1}})/\| \mathbf{A_{t}^{1}}-\mathbf{B_{t}^{1}}\|\), \(\mathbf{A_{t}^{1}}=(\mathbf{A_{t}^{1}}-\mathbf{C})\cdot Rot(\mathbf{e_{CB}}, \theta_{A})+\mathbf{C}\), \(\mathbf{B_{t}^{1}}=(\mathbf{B}-\mathbf{C})\cdot Rot(\mathbf{e_{A_{t}^{1}C^{ \prime}}},\varphi_{B})+\mathbf{C}\). Replacing vector \(\mathbf{P}\) in Equation (13) with the vertex position vector \(\mathbf{Q}\) of the mirror unit gives the position vector \(\mathbf{Q_{t}^{1}}\) after ThreeR-OneT motion and is defined by:  
\[\mathbf{Q_{t}^{1}}=(\mathbf{Q}-\mathbf{C})\cdot\mathbf{H}^{\text{ABC}}+ \mathbf{C_{t}^{1}}+\mathbf{M}_{1} \tag{14}\]  
Figure 4: The three rotational motions of the mirror unit in order ABC consist of: (**a**) rotation about axis CB, (**b**) rotation about axis A\({}_{\text{t}}^{1}\)C, and (**c**) rotation about axis B\({}_{\text{t}}^{1}\)A\({}_{\text{t}}^{1}\).