## 3 Simulations
### Simulation Settings and Results
The total number of simulated cases was 29,988. In all simulated cases, the constraints were satisfied and the solution existed. In 28,798 cases, the solver stopped on the condition on the step tolerance, while, in 1190 cases, it stopped on the optimality tolerance. The maximum number of iterations was never met. The values of  
\[\Delta\Omega =\Delta\Omega(\Omega_{1},\
u_{1},\Lambda,r)\] \[\Delta\
u =\Delta\
u(\Omega_{1},\
u_{1},\Lambda,r) \tag{48}\] \[\gamma_{1,2} =\gamma_{1,2}(\Omega_{1},\
u_{1},\Lambda,r)\]  
are the main results of this study. \(\Delta\Omega\) and \(\Delta\
u\) are the optima, while \(\gamma_{1,2}\) is the spacing angle between the satellites with respect to the emitter. Since they are 4D functions, the number of necessary graphical representations on the 2D plane for each function was 6. To take into account that in some representation planes, the results are in different ranges with respect to \(\
u_{1}\) because a shift in the latitude of the emitter requires a value of \(\
u_{1,0}=\Lambda\) for Nadir alignment, the independent variable \(\
u_{1}\) was replaced with \(\
u_{1}^{\Lambda}=(\
u_{1}-\
u_{1,0})=(\
u_{1}-\Lambda)\), as shown in (49)-(51). \[\Delta\Omega=\Delta\Omega(\Omega_{1},\
u_{1}-\
u_{1,0},\Lambda,r)=\Delta\Omega (\Omega_{1},\
u_{1}-\Lambda,\Lambda,r)=\Delta\Omega\Big{(}\Omega_{1},\
u_{1} ^{\Lambda},\Lambda,r\Big{)} \tag{49}\]  
\[\Delta\
u=\Delta\
u(\Omega_{1},\
u_{1}-\
u_{1,0},\Lambda,r)=\Delta\
u(\Omega _{1},\
u_{1}-\Lambda,\Lambda,r)=\Delta\
u\Big{(}\Omega_{1},\
u_{1}^{\Lambda},\Lambda,r\Big{)} \tag{50}\]  
\[\gamma_{1,2}=\gamma_{1,2}(\Omega_{1},\
u_{1}-\
u_{1,0},\Lambda,r)=\gamma_{1,2 }(\Omega_{1},\
u_{1}-\Lambda,\Lambda,r)=\Delta\gamma_{1,2}\Big{(}\Omega_{1}, \
u_{1}^{\Lambda},\Lambda,r\Big{)} \tag{51}\]  
In Figures 2-4, the dots are the calculated values obtained by the solver. Among them, the trend lines were obtained through an interpolation with a Piecewise Cubic Hermite  
Figure 1: Representation of the values of LOS constraints. (**a**) Values of the absolute constraint angle \(\delta\) with respect to the orbit height of the satellites \(h_{\mathrm{S}}\). (**b**) Values of the relative constraint angle \(\delta_{\mathrm{r}}\) with respect to the orbit height of the satellites \(h_{\mathrm{S}}\). Interpolating Polynomial of 100 points for each curve [42]: the choice of this interpolation method was made for graphical purposes and was due to its shape-preserving properties and to avoid as much as possible unwanted oscillations among the values. In particular, for the function \(\Delta\Omega=\Delta\Omega\big{(}\Omega_{1},\
u_{1}^{\Lambda},\Lambda,r\big{)}\), the following functions  
\[ \begin{split}\Delta\Omega&=\Delta\Omega\Big{(} \Omega_{1},\
u_{1}^{\Lambda},\Lambda_{*},r_{*}\Big{)}\\ \Delta\Omega&=\Delta\Omega\Big{(}\Omega_{1},\
u_{1,*}^{\Lambda},\Lambda,r_{*}\Big{)}\\ \Delta\Omega&=\Delta\Omega\Big{(}\Omega_{1,*},\
u_{1,*}^{\Lambda},\Lambda,r\Big{)}\\ \Delta\Omega&=\Delta\Omega\Big{(}\Omega_{1,*},\
u_{1,*}^{\Lambda},\Lambda,r\Big{)}\\ \end{split}  \tag{52}\]  
were considered for the representations in Figure 2, where \(\Omega_{1,*},\
u_{1,*}^{\Lambda},\Lambda_{*}\), and \(r_{*}\) are fixed values. Figure 2: _Cont_. For representations of trends, \(\Lambda_{*}=0^{\circ}\) was chosen, and \(r_{*}\) corresponded to the height \(h_{\mathrm{S_{*}}}=400\) km. Because of a jump discontinuity in the resulting trends for \(\Omega_{1}=0\) and \(\
u_{1}^{\Lambda}=0\), these values were not chosen for \(\Omega_{1,*}\) and \(\
u_{1,*}^{\Lambda}\). Instead, for \(\Omega_{1,*}\) and \(\
u_{1,*}^{\Lambda}\), the first positive values in the range of simulated values were considered. These first positive values depended on how the ranges for \(\Omega_{1}\) and \(\
u_{1}^{\Lambda}\) were defined and spaced in the simulation environment; in this study, 14 evenly spaced values were considered, but the ranges depended on the relative condition \(\delta_{r}\).