## 3 Spherical Area Coordinates (SACs)
### Local SACs
#### 3.2.1 Definition of SACs on the Spherical Triangle  
In the context of SACs, the reference triangle is the spherical triangle \(T\). The reference triangle can be any non-degenerate spherical triangle. Any area coordinates of the point in this reference triangle can be calculated through the vertices of this triangle. If there is a point \(\mathbf{p}\) inside \(T\), then we can divide \(T\) into three sub-triangles by connecting \(\mathbf{p}\) and the vertices of the reference triangle \(\mathbf{p}_{i},i=1,2,3\) with great circular arcs \(C_{i},i=1,2,3\) (see Figure 5(a)). **Definition 1**. : _A system of spherical area coordinates over a geodesic triangle \(T\) on \(\mathbb{S}^{2}\) is a one-to-one correspondence between points \(\mathbf{p}\in T\) and ordered triples of real numbers \(\mathbf{u}=(\lambda_{1},\lambda_{2},\lambda_{3})\) such that_  
1. \(0\leq\lambda_{i}\leq 1\)_;_ \(i=1,2,3\)_;_
2. \(\sum_{i}\lambda_{i}=1\)_;_
3. \(\lambda_{i}=area(T_{i})/area(T);i=1,2,3\)_,_  
_where \(T_{1}=(\mathbf{p},\mathbf{p}_{2},\mathbf{p}_{3})\),\(T_{2}=(\mathbf{p}_{1},\mathbf{p},\mathbf{p}_{3})\),\(T_{3}=(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p})\). It is obvious that the sub-triangles are all spherical triangles. Until this step, the coordinates of point \(\mathbf{p}\) can be written in the planar form (Equation (1)), but the issue lies in the method for calculating the areas of these four spherical triangles. Let a spherical triangle be drawn on the surface of a sphere of radius \(R\), centered at a point \(O=(0,0,0)\) (in Cartesian coordinates), with vertices \(P_{1}\), \(P_{2}\), and \(P_{3}\). Therefore, the vector from the center of the sphere to the vertices is given by \(\mathbf{p}_{1}=\widehat{O}\hat{P}_{1}\),\(\mathbf{p}_{2}=\widehat{O}\hat{P}_{2}\), and \(\mathbf{p}_{3}=\widehat{O}\hat{P}_{3}\). We assume that the area of the spherical triangle \(\widehat{\triangle}\) is \(S\), with \(R=1\). According to [PERSON]'s theorem in spherical trigonometry, the area of a spherical triangle \(\widehat{\triangle}\) on a sphere of radius \(R\) is_  
\[S=R^{2}(\alpha+\beta+\gamma-\pi), \tag{2}\]  
_where \(\alpha\), \(\beta\), and \(\gamma\) denote the angles between the sides (dihedral angles) in radians (shown in Figure 5(b)), and the angle \(\alpha\) is opposite the side that subtends the greater circle arc \(C_{1}\), and so forth. Hence, the SACs can be written as_  
\[ \begin{split}\lambda_{1}&=\frac{S_{1}}{S}=\frac{ \alpha_{T_{1}}+\beta_{T_{1}}+\gamma_{T_{1}}-\pi}{\alpha_{T}+\beta_{T}+\gamma_{T }-\pi},\\ \lambda_{2}&=\frac{S_{2}}{S}=\frac{\alpha_{T_{2}}+ \beta_{T_{2}}+\gamma_{T_{2}}-\pi}{\alpha_{T}+\beta_{T}+\gamma_{T}-\pi},\\ \lambda_{3}&=\frac{S_{3}}{S}=\frac{\alpha_{T_{3}}+ \beta_{T_{3}}+\gamma_{T_{3}}-\pi}{\alpha_{T}+\beta_{T}+\gamma_{T}-\pi}.\end{split}  \tag{3}\]  
Figure 6: SAC spherical reference triangle. When the spherical reference triangle \(T\) is determined, the area of \(T\) is a constant, and \(S_{i}\), \(i=1,2,3\), is a continuous function about the spherical point \(\mathbf{p}\). According to cosine laws of spherical trigonometry,  
\[ \begin{split}\alpha&=\cos^{-1}\left(\frac{\cos C_{1} -\cos C_{2}\cos C_{3}}{\sin C_{2}\sin C_{3}}\right);\\ \beta&=\cos^{-1}\left(\frac{\cos C_{2}-\cos C_{1} \cos C_{3}}{\sin C_{1}\sin C_{3}}\right);\\ \gamma&=\cos^{-1}\left(\frac{\cos C_{3}-\cos C_{2} \cos C_{1}}{\sin C_{2}\sin C_{1}}\right).\end{split}  \tag{4}\]  
The definitions of the dot product and cross-product are  
\[ \begin{split}\cos C_{1}&=\mathbf{p}_{2}\cdot\mathbf{p}_{3} ;\quad\cos C_{2}=\mathbf{p}_{1}\cdot\mathbf{p}_{3};\quad\cos C_{3}=\mathbf{p}_{2}\cdot\mathbf{ p}_{1};\\ \sin C_{1}&=\|\mathbf{p}_{2}\times\mathbf{p}_{3}\|;\quad \sin C_{2}=\|\mathbf{p}_{3}\times\mathbf{p}_{1}\|;\quad\sin C_{3}=\|\mathbf{p}_{1}\times \mathbf{p}_{2}\|.\end{split} \]  
Note that \(\|\mathbf{p}\|=\|\mathbf{p}_{1}\|=\|\mathbf{p}_{2}\|=\|\mathbf{p}_{2}\|=1\). Then, we obtain the formula below. \[