## 3 Optimization Arrangement of the One-Camera-Five-Mirror Module
### Optimization Variables  
The four length optimization variables are defined as follows: \(l_{OA}=a>0\), \(l_{OB}=b>0\), \(l_{BC}=c>0\), \(l_{CH}=d>0\). The three angle optimization variables are defined as follows \(\theta_{1}\) (mirror A), \(\theta_{2}\) (mirror B) and \(\theta_{3}\) (mirror C). Hence, the optimal objective can be represented as \(l_{OP}=l_{OB}=b+c+d\) and the coordinates of the virtual cameras 2 and 3 are expressed as follows: \(Q(\frac{b+c+d}{\sqrt{2}},-\frac{b+c+d}{\sqrt{2}})\), \(P(\frac{b+c+d}{\sqrt{2}},\frac{b+c+d}{\sqrt{2}})\). In addition, the coordinates of the plane mirrors A and B can be expressed as \(A(\frac{a}{\sqrt{2}},\frac{a}{\sqrt{2}})\), \(B(\frac{b}{\sqrt{2}},-\frac{b}{\sqrt{2}})\).  
All the coordinates in Figure 4 can be deviated and expressed by the optimization variables, \(a,b,c,d,\theta_{1},\theta_{2},\theta_{3}\), based on the optical and geometrical relationship among them. In addition, the coordinate values are listed in Table 1.  
Figure 4: Establishment and parameter setting of the optimization model.  
The calculation of the following parameters was to define the constraint conditions to meet the requirements of the one-camera-five-mirror 3D imaging module and ensure no interference between the optical paths.  
To prevent the optical image reflected by the mirror A from being interrupted by the valve, there should be a certain interval between \(A_{0}\) and \(V_{0}\), as expressed in Equation (2):  
\[x_{A_{0}}-x_{V_{0}}>3. \tag{2}\]  
The left boundary of the optical path reflected by the mirror C is on the positive side of the \(y\)-axis, in case of influencing the mirror belonging to the virtual camera 1, which is symmetrical with the mirror C. Thus, the \(x\)-coordinate should meet the following constraint:  
\[x_{C_{1}}>1.\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Point** & \(x\) **coordinate** & \(y\) **coordinate** \\ \hline H & \(x_{p}+y_{p}\tan\theta_{1}-y_{H}\tan\theta_{1}\) & \(\dfrac{2x_{p}\tan\theta_{1}+y_{p}(\tan^{2}\theta_{1}-1)+\sqrt{2}a(1-\tan\theta _{1})}{1+\tan^{2}\theta_{1}}\) \\ R & \(x_{Q}+y_{Q}\tan\theta_{2}-y_{R}\tan\theta_{2}\) & \(\dfrac{2x_{Q}\tan\theta_{2}+y_{Q}(\tan^{2}\theta_{2}-1)-\sqrt{2}b(1+\tan\theta _{2})}{1+\tan^{2}\theta_{2}}\) \\ C & \(\dfrac{\frac{y_{p}-y_{p}}{x_{B}-x_{B}}x_{R}-y_{R}-\frac{x_{B}+y_{H}}{2}\tan \theta_{3}+\frac{y_{R}+y_{H}}{2}}{\frac{y_{B}-y_{R}}{x_{B}-x_{R}}-\tan\theta_{ 3}}\) & \(\dfrac{y_{B}-y_{R}}{x_{B}-x_{R}}(x_{C}-x_{R})+y_{R}\) \\ \(V_{1}\) & \(r\cos(\arccos\dfrac{r}{\sqrt{x_{p}^{2}+y_{p}^{2}}}+45^{\circ})\) & \(r\sin(\arccos\dfrac{r}{\sqrt{x_{p}^{2}+y_{p}^{2}}}+45^{\circ})\) \\ \(A_{1}\) & \(\dfrac{y_{p}-\frac{y_{V_{1}}-y_{p}}{x_{V_{1}}-x_{p}}x_{p}-\frac{a}{\sqrt{2}}(1 -\tan\theta_{1})}{\tan\theta_{1}-\frac{y_{V_{1}}-y_{p}}{x_{V_{1}}-x_{p}}}\) & \(x_{A_{1}}\tan\theta_{1}+\dfrac{a}{\sqrt{2}}(1-\tan\theta_{1})\) \\ \(A_{0}\) & \(x_{H}-\dfrac{x_{A_{1}}-x_{H}}{y_{A_{1}}-y_{H}}y_{H}\) & \(0\) \\ \(V_{0}\) & \(\dfrac{ly}{\sqrt{2}}\) & \(0\) \\ \(V_{2}\) & \(x_{V_{1}}\) & \(-y_{V_{1}}\) \\ \(B_{1}\) & \(\dfrac{\frac{y_{Q}-y_{N_{2}}}{x_{Q}-x_{V_{2}}}x_{Q}-y_{Q}-\frac{b}{\sqrt{2}}(1 +\tan\theta_{2})}{\frac{y_{Q}-y_{N_{2}}}{x_{Q}-x_{V_{2}}}-\tan\theta_{2}}\) & \(x_{B_{1}}\tan\theta_{2}-\dfrac{b}{\sqrt{2}}(1+\tan\theta_{2})\) \\ \(C_{1}\) & \(\dfrac{\frac{y_{R}+y_The slope of the line \(HC_{2}\) and \(HV_{0}\) can be expressed as:  
\[k_{HC_{2}} =\frac{y_{H}-y_{C_{2}}}{x_{H}-x_{C_{2}}},\] \[k_{HV_{0}} =\frac{y_{H}-y_{V_{0}}}{x_{H}-x_{V_{0}}}.\]  
To prevent the optical path from the mirror A and C from interfering with each other, the slopes angle of the line \(HC_{2}\) and \(HA_{1}\) should meet:  
\[\arctan k_{HC_{2}}-\arctan k_{HV_{0}}>\frac{1^{\circ}}{180^{\circ}}\pi. \tag{4}\]