## 4 The Global SAC and Grid Systems
### The Octaledron Version of the Global Grid System  
We can also define SACs on the octahedron. First, we need to orient the coordinate sphere triangles. The octahedron is oriented in the following way: the center of the sphere is at the \((x,y,z)\) coordinates \((0,0,0)\), and the six vertices are at \(V_{1}=(1,0,0)\), \(V_{2}=(0,1,0)\), \(V_{3}=(0,0,1)\), \(V_{4}=-V_{1}\), \(V_{5}=-V_{2}\), and \(V_{6}=-V_{3}\) (shown in Figure 11).  
Then, we divide the sphere into eight spherical triangles. We denote these regions as \(N_{i}\) (north) and \(S_{i}\) (south), \(i=1\ldots 4\). Therefore, any point on the sphere surface must be in one of these eight cells, and the cells can serve as the reference triangle. Because the derivation is the same as the tetrahedron version, the forward transformation laws have the following forms:  
\[\begin{split}\lambda_{1}^{\prime}&=\frac{4}{\pi} \tan^{-1}\left(\frac{|x|}{1+|y|+|z|}\right),\\ \lambda_{2}^{\prime}&=\frac{4}{\pi}\tan^{-1}\left( \frac{|y|}{1+|x|+|z|}\right),\\ \lambda_{3}^{\prime}&=\frac{4}{\pi}\tan^{-1}\left( \frac{|z|}{1+|x|+|y|}\right).\end{split} \tag{13}\]  
Different from the tetrahedron version of SACs, the octahedron version of SACs have three components \((\lambda_{1},\lambda_{2},\lambda_{3})\). In Equation (13), \(\lambda_{1}^{\prime}\), \(\lambda_{2}^{\prime}\), and \(\lambda_{3}^{\prime}\) are all positive. The SACs in different regions have different sign patterns and are determined by the regions containing the point \(P\). The sign patterns are the same as the octants in the 3D Cartesian system (illustrated in Figure 12).  
Figure 11: The octahedron version of the initial global grid.  
Figure 10: The 4-fold tetrahedron version Discrete Global Grid Systems (DGGSs) based on SACs (level = 2, 3, 4).  
The inverse transformations are as follows:  
\[x^{\prime}=\frac{-M_{\lambda 1}}{F},\quad y^{\prime}=\frac{-M_{\lambda 2}}{F}, \quad z^{\prime}=\frac{-M_{\lambda 3}}{F}, \tag{14}\]  
where \(M_{\lambda 1}\), \(M_{\lambda 2}\), \(M_{\lambda 3}\), and \(F\) are third-order determinants:  
\[F=\begin{vmatrix}-\cos(u)&\sin(u)&\sin(u)\\ \sin(v)&-\cos(v)&\sin(v)\\ \sin(w)&\sin(w)&-\cos(w)\end{vmatrix},\quad M_{\lambda 1}=\begin{vmatrix}\sin(u)& \sin(u)\\ \sin(v)&-\cos(v)&\sin(v)\\ \sin(w)&\sin(w)&-\cos(w)\end{vmatrix},\]  
\[M_{\lambda 2}=\begin{vmatrix}-\cos(u)&\sin(u)&\sin(u)\\ \sin(v)&\sin(v)&\sin(v)\\ \sin(w)&\sin(w)&-\cos(w)\end{vmatrix},\quad M_{\lambda 3}=\begin{vmatrix}-\cos(u)& \sin(u)&\sin(u)\\ \sin(v)&-\cos(v)&\sin(v)\\ \sin(w)&\sin(w)&\sin(w)\end{vmatrix},\]  
where \(u=\pi|\lambda_{1}|/4\), \(v=\pi|\lambda_{2}|/4\), and \(w=\pi|\lambda_{3}|/4\). Finally, according to the regions to which the point belongs, we assign the same sign patterns to the Cartesian coordinates \((x,y,z)\):  
\[\mathbf{p}\in N_{0} \Leftrightarrow(x=x^{\prime},y=y^{\prime},z=z^{\prime});\] \[\mathbf{p}\in N_{1} \Leftrightarrow(x=-x^{\prime},y=y^{\prime},z=z^{\prime});\] \[\mathbf{p}\in N_{2} \Leftrightarrow(x=-x^{\prime},y=-y^{\prime},z=z^{\prime});\]  
and so forth. The main advantages of the octahedron version of SACs are that it has more symmetric coordinates than the tetrahedron version, and all regions share a unique equation. Figure 13 illustrates the octahedron version 4-fold DGGSs at levels 1, 2, 3, and 4.  
Figure 12: The sign patterns of the octahedron version of SACs.  
Figure 13: Four-fold DGGSs based on the octahedron version of SACs (level = 1, 2, 3, 4).