## 4 Method to Represent an Arbitrary Observation Geometry  
The classical orbital parameters can be expressed as \((a,\,e,\,i,\,\Phi,\,\,\omega,\,M)\). \(a\) is the semi-major axis, \(e\) is the eccentricity, \(i\) is the inclination, \(\Phi\) is the right ascension of ascending node, \(w\) is the argument of perigee, and \(M\) is the mean anomaly. The classical orbit elements of the optical sensor orbit are \((a_{\text{Sensor}}\), \(0\), \(i\), sensor, \(\Phi_{\text{Sensor}}\), \(\omega_{\text{Sensor}}\), \(M_{\text{Sensor}}\)), and those of the target orbit are \(\left(a_{\text{Target}}\right.\), \(0\), \(i_{\text{Target}}\), \(\Phi_{\text{Target}}\), \(\omega_{\text{Target}}\), \(M_{\text{Target}}\)). The angular momentum vector \(L_{\text{Target}}\) of the target can be defined as the direction of the \(x\)-axis, and \(L_{\text{Target}}\times L_{\text{Sensor}}\) as the direction of the \(z\)-axis. The direction of the \(y\)-axis can be determined according to the right-hand rule. In the new coordinate system, the classical orbit elements of the optical sensor orbit are \(\left(a_{\text{Sensor}}\right.\), \(0\), \(i\), \(0^{\circ}\), \(0^{\circ}\), \(M_{\text{Sensor}}^{\prime}\)), and those of the target orbit are \(\left(a_{\text{Target}}\right.\), \(0\), \(0^{\circ}\), \(0^{\circ}\), \(0^{\circ}\), \(M_{\text{Target}}^{\prime}\)). There exists a relationship between vector \(\mathbf{r}^{\prime}\) in the new coordinate system and vector \(\mathbf{r}\) in the natural coordinate system:  
\[\mathbf{r}^{\prime}=R_{0}R_{TA_{\text{Target}}}R_{\omega_{\text{Target}}}R_{i_{ \text{Target}}}R_{\Phi_{\text{Target}}}\mathbf{r} \tag{13}\]  
where \(TA_{\text{Target}}\) is the true anomaly of the target. \(R_{TA_{\text{Target}}}\), \(R_{\omega_{\text{Target}}}\), \(R_{i_{\text{Target}}}\), and \(R_{\Phi_{\text{Target}}}\) can be expressed as:  
\[R_{TA_{\text{Target}}}=\left[ \begin{array}{ccc}\cos\left(TA_{\text{Target }}\right)&\sin\left(TA_{\text{Target}}\right)&0\\ -\sin\left(TA_{\text{Target}}\right)&\cos\left(TA_{\text{Target}}\right)&0\\ 0&0&1\end{array} \right] \tag{14}\] \[R_{\omega_{\text{Target}}}=\left[ \begin{array}{ccc}\cos\left( \omega_{\text{Target}}\right)&\sin\left(\omega_{\text{Target}}\right)&0\\ -\sin\left(\omega_{\text{Target}}\right)&\cos\left(\omega_{\text{Target}} \right)&0\\ 0&0&1\end{array} \right]\] \[R_{i_{\text{Target}}}=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos\left(i_{\text{Target}}\right)&\sin\left(i_{\text{Target}}\right)\\ 0&-\sin\left(i_{\text{Target}}\right)&\cos\left(i_{\text{Target}}\right) \end{array} \right]\] \[R_{\Phi_{\text{Target}}}=\left[ \begin{array}{ccc}\cos\left( \Phi_{\text{Target}}\right)&\sin\left(\Phi_{\text{Target}}\right)&0\\ -\sin\left(\Phi_{\text{Target}}\right)&\cos\left(\Phi_{\text{Target}}\right)&0 \\ 0&0&1\end{array} \right]\]  
\(R_{0}\) can be expressed as:  
\[R_{0}=\left[ \begin{array}{ccc}L_{x}(1)&L_{x}(2)&0\\ -L_{x}(2)&L_{x}(1)&0\\ 0&0&1\end{array} \right] \tag{15}\]  
where:  
\[L_{x}{}^{\prime}=\left(R_{TA_{\text{Target}}}R_{\omega_{\text{Target}}}R_{i_{ \text{Target}}}R_{\Phi_{\text{Target}}}\right)^{-1}L_{\text{Target}}\times L_{ \text{Sensor}} \tag{16}\]  
Thus, an optical sensor with orbital parameters \(\left(a_{\text{Sensor}},0,i,0^{\circ},0^{\circ},M_{\text{Sensor}}{}^{\prime}\right)\), observing a target with orbital parameters \(\left(a_{\text{Target}},0,0^{\circ},0^{\circ},0^{\circ},M_{\text{Target}}{}^{ \prime}\right)\), can represent an arbitrary observation geometry. It is worth noting that, when the optical sensor and target orbit parameters are \(\left(a_{\text{Sensor}},0,i,0^{\circ},0^{\circ},0\right)\) and \(\left(a_{\text{Target}},0,0^{\circ},0^{\circ},0^{\circ},0\right)\), the optical sensor and target are aligned with the centre of the Earth. \(i\) corresponds to the angle between the target and the optical sensor orbital plane, where \(0<i<\pi\). \(i>\frac{\pi}{2}\) implies that the optical sensor is moving in the opposite direction to the target.