## 2 Optical Cone and Pendulum Scanning Imaging Method
Then, from the pixel on the image plane coordinate frame to the target point in the geographic coordinate frame can be represented by:  
\[ \begin{bmatrix}X_{G}\\ Y_{G}\\ Z_{G}\\ 1\end{bmatrix} =C_{s}^{G} \begin{bmatrix}X_{s}\\ Y_{s}\\ Z_{s}\\ 1\end{bmatrix} =C_{s}^{G}C_{c}^{s} \begin{bmatrix}X_{c}\\ Y_{c}\\ Z_{c}\\ 1\end{bmatrix} =C_{s}^{G}C_{c}^{s}C_{p}^{c} \begin{bmatrix}X_{p}\\ Y_{p}\\ Z_{p}\\ 1\end{bmatrix}  \tag{1}\]  
where the image plane coordinate frame needs to translate \(f\) along the positive direction of the \(z_{p}\) axis to the camera coordinate frame, and the transformation matrix is \(C_{p}^{c}\). From the camera coordinate frame to the satellite coordinate frame, it needs to rotate the angle \(\gamma\) around the \(y_{c}\) axis and enlarges it by \(\frac{H}{f}\) times, and its transformation matrix is \(C_{c}^{s}\). From the satellite coordinate frame to the geographic coordinate frame, it needs to translate \(H\) along the negative direction of the \(z_{s}\) axis and rotate \(180^{\circ}\) around the \(x_{s}\) axis, and the change matrix is \(C_{s}^{G}\). The three transformation matrices satisfy  
\[C_{p}^{c}= \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&f\\ 0&0&0&1\end{bmatrix} \]  
\[C_{c}^{s}= \begin{bmatrix}\frac{H}{f}&0&0&0\\ 0&\frac{H}{f}&0&0\\ 0&0&\frac{H}{f}&0\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}cos\gamma&0&-sin\gamma&0\\ 0&1&0&0\\ sin\gamma&0&cos\gamma&0\\ 0&0&0&1\end{bmatrix} \]  
\[C_{s}^{G}= \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&-H\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}1&0&0&0\\ 0&cos\tau&sin\tau&0\\ 0&-sin\tau&cos\tau&0\\ 0&0&0&1\end{bmatrix} \]  
where \(f\) is the camera focal length. \(H\) is the satellite orbital altitude. \(\gamma\) is the angle between the camera optical axis and the nadir axis, including the initial tilt angle \(\alpha\) and the camera swing angle \(\beta\). Then, the projection of the image plane on the Earth can be calculated recursively according to \([X_{G},\,Y_{G},\,Z_{G}]^{T}\), and the position of the satellite on the Earth-centered Earth-fixed (ECEF) coordinate system. Equation (1) will be used in subsequent simulations. Next, we design the basic parameters through geometric analysis, as shown in Figures 5 and 6. First, \(FOV\) is the field of view of the camera, which is determined by the detector surface size \(L\) and the focal length \(f\) of the camera. The detector area size is determined by the pixel size \(a\) and the number of pixels \(n_{1}\times n_{2}\). Then, \(FOV\) can be expressed as  
\[FOV=2 arctan\left(\frac{L}{2f}\right)=2\text{arctan}\left(\frac{a\times n_{1}} {2f}\right) \tag{2}\]As shown in Figure 6, according to the user's requirements for ground coverage width and ground resolution, the camera parameters such as the image plane size \(L\), field of view angle \(FOV\), focal length \(f\), and satellite parameters such as satellite height \(H\) and camera tilt angle \(\gamma\) are determined. Different from traditional imaging with a small field of view, under the condition of ultra-large width of thousand kilometers, the ground needs to be  
Figure 5: Mapping of the image plane to the Earth. Figure 6: The actual mapping at a large width. regarded as a sphere.