## 3 ABP Determination Models and BPE Computing Method
### Ideal ABP Determination Model  
For optical sensors, the traditional MO method is used to determine the pointing vector (PV) of each image pixel in the ICS [20; 30]. The key aspect of this method is to use the 3-by-3 reflection matrix (RM) of each optical mirror to describe the propagation path of the optical ray from the ground location to the corresponding pixel location in the ICS. Since we were only concerned with the pointing information of the antenna beam rather than other antenna performance indicators, it was reasonable to adopt the MO method for rapid ABP determination. However, it must be pointed out that the ABP determination is quite different from the pointing determination of the optical ray. Compared with optical sensors, space-borne microwave antennas include not only planar reflectors but also curved reflectors. For a planar reflector, the outward-pointing normal direction is the same for arbitrary points on the surface. Thus, the translation of the incident beam on the planar reflector does not affect the pointing of the reflected beam, while for a curved reflector, the outward-pointing normals are in different directions at different points on the surface. As a result, a small drift in the point of emergence (POE) of the beam will result in a relatively large deviation in the pointing of the reflected beam. Therefore, the key point to developing the ABP determination model is to trace the beam's PV and POE simultaneously as it propagates in the reflector system. To describe the propagation of the antenna beam in a more unified and compact form, we extended the traditional 3-D PV to a 7-D generalized pointing vector (GPV):  
\[\mathbf{L}=\left[ \begin{array}{cccccc}\mathbf{I}&\mathbf{r}&\mathbf{1}\end{array} \right]^{ \rm T}=\left[ \begin{array}{cccccc}l_{x}&l_{y}&l_{z}&r_{x}&r_{y}&r_{z}&1\end{array}  \right]^{\rm T} \tag{1}\]  
where \(\mathbf{I}=\left[ \begin{array}{cccccc}l_{x}&l_{y}&l_{z}\end{array} \right]^{\rm T}\) is the traditional 3-D PV and \(\mathbf{r}=\left[ \begin{array}{cccccc}r_{x}&r_{y}&r_{z}\end{array} \right]^{\rm T}\) represents the coordinates of the POE of the beam in a specific coordinate system. To be compatible with the 7-D GPV, we further extended the traditional 3-by-3 attitude transformation matrix (ATM) and RM to a 7-by-7 generalized transformation matrix (GTM) \(\mathbf{M}_{i}^{i}\) and generalized reflection matrix (GRM) \(\mathbf{F}_{i}\), respectively. The explicit expressions for \(\mathbf{M}_{i}^{i}\) and \(\mathbf{F}_{i}\) are as follows:  
\[\mathbf{M}_{i}^{i}=\left[ \begin{array}{cccccc}\mathbf{Rot}_{i}^{i}&\mathbf{O}_{3\times 3 }&\mathbf{O}_{3\times 1}\\ \mathbf{O}_{3\times 3}&\mathbf{Rot}_{i}^{i}&\mathbf{I}_{j}\\ \mathbf{O}_{1\times 3}&\mathbf{O}_{1\times 3}&1\end{array} \right] \tag{2}\]  
\[\left\{ \begin{array}{ll}\mathbf{F}_{i}=\left[\begin{array}{cccccc}\mathbf{Ref}_{i} ^{i}&\mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 1}\\ k_{i}\mathbf{E}_{3\times 3}&\mathbf{E}_{3\times 3}&\mathbf{O}_{3\times 1}\\ \mathbf{O}_{1\times 3}&\mathbf{O}_{1\times 3}&1\end{array}\right]\\ \mathbf{Ref}_{i}=\left[\begin{array}{cccccc}1-2n_{x}^{2}&-2n_{x}n_{y}&-2n_{x}n_ {z}\\ -2n_{x}n_{y}&1-2n_{y}^{2}&-2n_{y}n_{z}\\ 2n_{x}n_{z}&-2n_{y}n_{z}&1-2n_{z}^{2}\end{array}\right]\end{array} \right. \tag{3}\]where \(\mathbf{Rot}_{i}^{i}\) is the 3-by-3 ATM which transforms the pointing vectors between representations from coordinate system \(i\) (whose origin is denoted as \(I\)) to coordinate system \(j\) (whose origin is denoted as \(J\)), \(\mathbf{I}_{j}\) is the \(j\) coordinate representation of the origin \(I\), \(\mathbf{Ref}_{i}\) is the 3-by-3 RM at a certain point in coordinate system \(i\), \(\left[ \begin{array}{cccc}n_{x}&n_{y}&n_{z}\end{array} \right]^{\rm T}\left[ \begin{array}{cccc}n_{x}&n_{y}&n_{z}\end{array} \right]^{\rm T}\) is the outward-pointing normal to a reflector at this particular point; \(k_{i}\) is the optical path length of the incident beam, and \(\mathbf{O}\) and \(\mathbf{E}\) represent the zero matrix (vector) and identity matrix, respectively.