## 3 Control Algorithm Description
### Reference Trajectories  
The free motion trajectory (2) can be rewritten as follows:  
\[\left\{ \begin{array}{l}x_{ij}(t)=-3B_{1}^{ij}\omega t+2B_{2}^{ij}\cos(\omega t +\alpha_{ij})+B_{4}^{ij},\\ y_{ij}(t)=B_{3}\sin(\omega t+\beta_{ij}),\\ z_{ij}(t)=2B_{1}^{ij}+B_{2}^{ij}\sin(\omega t+\alpha_{ij}),\end{array} \right. \tag{10}\]  
where \(B_{1}^{ij}-B_{4}^{ij}\) are trajectory constants defined using \(C_{1}^{ij}-C_{6}^{ij}\) constants from Equation (3) as follows:  
\[B_{1}^{ij}=C_{1}^{ij},\ B_{2}^{ij}=\sqrt{\left(C_{2}^{ij}\right)^{2}+\left(C_ {3}^{ij}\right)^{2}},\ B_{3}^{ij}=\sqrt{\left(C_{5}^{ij}\right)^{2}+\left(C_{6 }^{ij}\right)^{2}},\ B_{4}^{ij}=C_{4}^{ij}, \tag{11}\]  
\(\alpha_{ij},\beta_{ij}\) are phase angles. If the relative drift parameter \(B_{1}^{ij}\) and the relative trajectory shift parameter \(B_{4}^{ij}\) equal zero, and the in-plane amplitude \(B_{2}^{ij}\) and the out-of-plane amplitude \(B_{3}^{ij}\) satisfy the relation \(B_{3}^{ij}=2B_{2}^{ij}\), then the relative trajectories are projected circular orbits. It means that under these initial conditions for any time \(t\), the following relation holds true for the trajectory coordinates:  
\[x_{ij}^{2}(t)+y_{ij}^{2}(t)=x_{ij}^{2}(0)+y_{ij}^{2}(0)=\left(B_{3}^{ij} \right)^{2}=const \tag{12}\]  
Projected circular relative trajectories are well-suited for image demonstration by sunlight reflectors because during the motion the relative geometry is maintained for the Earth observer, although during the flight the image will rotate with orbital angular velocity \(\omega\) relative to the image center. If all the satellites have the same out-of-plane phase angle \(\beta_{ij}\), then by choosing the in-plane phase angles \(\alpha_{ij}\) and the trajectory radius \(B_{3}^{ij}\), it is possible to set the required relative position of each satellite. Consider, for example, that some satellites are initially placed in two perpendicular lines, and they start to move along the projected circular reference orbits. Figure 3 presents the initial positions of nine satellites on perpendicular red and green lines and the closed relative trajectories in the \(Oxy\) plane. Two initially perpendicular lines formed by the satellites remain perpendicular, which means if the letters to be demonstrated are constructed in this way they will not be deformed for the observer at zenith. The projected circular trajectory in the \(Oxy\) plane is a linear approximation of a more complex natural periodic closed trajectory. A model to describe it with greater accuracy uses the solution to Hill-Clohessy-Wiltshire Equation (5) expressed in the curvilinear coordinates. The trajectory (5) can be rewritten into the form similar to Equation (10):  
\[\left\{ \begin{array}{l}\varphi_{ij}(t)=-3\widetilde{B}_{1}^{ij}\omega t+2 \widetilde{B}_{2}^{ij}\cos(\omega t+\widetilde{\alpha}_{ij})+\widetilde{B}_{4 }^{ij},\\ \theta_{ij}(t)=\widetilde{B}_{3}\sin(\omega t+\widetilde{\theta}_{ij}),\\ \rho_{ij}(t)=2\widetilde{B}_{1}^{ij}a_{0}+\widetilde{B}_{2}^{ij}a_{0}\sin( \omega t+\widetilde{\alpha}_{ij}),\end{array} \right. \tag{13}\]  
where \(\widetilde{B}_{1}^{ij}-\widetilde{B}_{4}^{ij},\widetilde{\alpha}_{ij}, \widetilde{\beta}_{ij}\) are the constants defined using \(\widetilde{C}_{1}^{ij}-\widetilde{C}_{6}^{ij}\) similarly to (11). The condition (12) for the projected circular orbit in the curvilinear coordinates becomes:  
\[\left(\varphi_{ij}(t)a_{0}\right)^{2}+\left(\theta_{ij}(t)a_{0}\right)^{2}= \left(\widetilde{B}_{3}^{ij}a_{0}\right)^{2}=const. \tag{14}\]  
The trajectory expressed in the curvilinear coordinates with zero relative drift parameter \(\widetilde{B}_{1}^{ij}\) and zero relative shift parameter \(\widetilde{B}_{4}^{ij}\) subject to condition (14) has circular projection on the sphere with radius \(a_{0}\), in contrast to circular projection on the \(Oxy\) plane for Cartesian coordinates. This kind of closed trajectory is also acceptable for the Earth observer, since at close relative distance the relative geometry is not notably affected.