## 2 EarM_EPnP
### Threshold Value of Central Angle  
In Section 2.2, this method solved the camera positions in two different coordinate systems. When the height of the camera was low in the sky, the two values \((X_{s},Y_{s},Z_{s})\) and \(\left(X_{f},Y_{f},Z_{f}\right)\) were infinitely close, because the radius of the Earth is very large. In the EarM_EPnP method, the circular surface of the sphere is approximated to the plane. However, in the process of increasing the camera height, the error is generated by the traditional EPnP method. As the height increases, the error will continue to increase. The relationship between the camera height and the center angle threshold is better solved, and maximum center angle thresholds with errors within 1 m, 10 m, and 100 m are obtained. In this method, the Earth was regarded as a standard sphere, and the radius was the average radius of the Earth. The EarM_EPnP proposed in this method becomes a true value under this model. \((X_{s},Y_{s},Z_{s})\) is the real position of the camera under the model. The following is a general formula derivation for calculating the relationship between the height and the central angle within a specific error range. Firstly, it was necessary to obtain the new coordinates of the solution point set \(A^{\prime}\) in the two-dimensional image coordinate system. Through the calculation of Sections 2.1 and 2.2, the solution point set \(A^{\prime}\), the camera's real position \((X_{s},Y_{s},Z_{s})\), the EpnP measurement position \(\left(X_{f},Y_{f},Z_{f}\right)\), and the camera's internal reference matrix \(A\) (the internal reference matrix of each camera is unique) were obtained. The intrinsic matrix \(A\) contains the effective focal length values \(f_{x},f_{y}\) and the principal point translation values \(u_{0},v_{0}\). The internal reference matrix \(A\) is shown in Equation (6). \[A= \begin{bmatrix}f_{x}&0&u_{0}\\ 0&f_{y}&v_{0}\\ 0&0&1\end{bmatrix}  \tag{6}\]  
Through the internal parameter matrix \(A\), the solution point set \(A^{\prime}\) in the spherical coordinate system could be mapped to the two-dimensional image coordinate system. The mapping formula is shown in Equations (7) and (8). \[X_{{}_{lang,n}}^{S}=A\times{A_{n}}^{\prime}=\left[ \begin{array}{l}R\sin\theta \cos\varphi\cdot f_{x}+R\cos\theta\cdot u_{0}\\ R\sin\theta\sin\varphi\cdot f_{y}+R\cos\theta\cdot v_{0}\\ R\cos\theta\end{array} \right] \tag{7}\]  
Equation (7) describes the preparation for generating two-dimensional image coordinates by multiplying the coordinates of the Earth's surface points and the internal reference matrix. In this method, each point in the calculated point set \(A^{\prime}\) is brought into Equation (7) for a solution, and a new point set \(X_{{}_{lang,n}}^{S}\) can be obtained. Each element in the point set is a matrix with a size of \(3\times 1\). The number of point sets is also \(n\). For subsequent calculations, this method normalizes each element \(X_{{}_{lang,n}}^{S}\) in the point set \(X_{{}_{lang}}^{S}\), and the normalization method is shown in Equation (8). \[ \begin{bmatrix}u&v&1\end{bmatrix} =X_{{}_{lang}}^{S}= \begin{bmatrix}\frac{R \sin\theta\cos\varphi\cdot f_{x}+R\cos\theta\cdot u_{0}}{R\cos\theta}\\ \frac{R\sin\theta\sin\varphi\cdot f_{x}+R\cos\theta\cdot v_{0}}{R\cos\theta} \\ 1\end{bmatrix} ^{T} \tag{8}\]  
In Equation (8), this method divides all the elements \(X_{{}_{lang}}^{S}\) by the third row of elements and then determines the transpose. At this point, the coordinate \(A^{\prime}\) of the point set \(X_{{}_{Alg,n}}^{S}\) in the two-dimensional image coordinate system is obtained using Equations (7) and (8). Secondly, to make the pose of the camera correspond to the position and orientation of the object in the real world, it was necessary to obtain the new coordinates of the solution point set \(A^{\prime}\) in the world coordinate system.