## 3 Theory
### Incidence Function
\tag{10}\]  
Using the trigonometric identity for the cosine of the sum and difference of the angles, terms are further simplified:  
\[\left[ \begin{array}{c}cos(\varphi-\varphi_{P}+\Delta\varphi)\\ -cos(\varphi-\varphi_{P}-\Delta\varphi)\end{array} \right]=-2\sin\ (\varphi-\varphi_{P})sin(\Delta\varphi), \tag{11}\]  
and  
\[\left[ \begin{array}{c}cos(\varphi-\varphi_{S}+\Delta\varphi)\\ -cos(\varphi-\varphi_{S}-\Delta\varphi)\end{array} \right]=-2\sin\ (\varphi- \varphi_{S})sin(\Delta\varphi). \tag{12}\]  
Substituting Equations (11) and (12) into Equations (10) and (11), respectively, differences in arguments may be expressed:  
\[\Phi_{S2}(\varphi_{S},\theta_{S})-\Phi_{S1}(\varphi_{S},\theta_{S})=2\ k\rho \ sin\theta_{S}\left[\sin\ (\varphi-\varphi_{S})sin(\Delta\varphi)\right]+\Delta\L\ \cos \theta_{S}, \tag{13}\]  
and  
\[\Phi_{P2}(\varphi_{P},\theta_{P})-\Phi_{P1}(\varphi_{P},\theta_{P})=2\ k\rho \ sin\theta_{P}\left[\sin\ (\varphi-\varphi_{P})sin(\Delta\varphi)\right]+\Delta\L\ \cos \theta_{P}. \tag{14}\]  
Substituting Equations (13) and (14) into Equation (4), the latter may be partially simplified:  
\[M_{SP}(\varphi_{S},\theta_{S};\varphi_{P},\theta_{P})=\left\{ \begin{array}[] {c}+M_{S}\{1+cos\{2k\rho\ sin\theta_{S}\left[\sin\ (\varphi-\varphi_{S})sin(\Delta\varphi)\right]+\Delta\L\ \cos \theta_{S}\}\}\\ +M_{P}\{1+cos\{2k\rho\ sin\theta_{P}\left[\sin\ (\varphi-\varphi_{P})sin(\Delta \varphi)\right]+\Delta\L\ \cos\theta_{P}\}\}\end{array} \right.. \tag{15}\]  
Terms \(\Delta Los\theta_{i}\) in each line represent the initial phase values, having different values for the star, \(\theta_{S}\), and planet, \(\theta_{P}\). The planet value slowly changes when the planet orbits the star during a local year. The star value remains unchanged during the observation time, for all practical purposes. They are set both to zero momentarily, because they change slowly during the observation time or the image capture. \[M_{SP}(\varphi_{S},\theta_{S};\varphi_{P},\theta_{P})=\left\{ \begin{array}{c} +M_{S}\{1+cos\{2k\rho\ sin\theta_{S}\left[\sin\ (\varphi-\varphi_{S})sin(\Delta\varphi)\right]\}\}\\ +M_{P}\{1+cos\{2k\rho\ sin\theta_{P}\left[\sin\ (\varphi-\varphi_{P})sin(\Delta \varphi)\right]\}\}\end{array} \right.. \tag{16}\]  
Equation (16) represents two fringe patterns. The argument of the cosine term in the first line is \(2k\rho\ sin\theta_{S}\left[\sin\ (\varphi-\varphi_{S})sin(\Delta\varphi)\right]\). The star fringes, therefore, lie along the line \(\varphi=\varphi_{S}\). That is, the fringe gray value is unchanged along this direction, meaning that the lines of equal gray value are aligned parallel to this direction. According to the argument of the cosine in the second line, \(2k\rho\ sin\theta_{P}\left[\sin\ (\varphi-\varphi_{P})sin(\Delta\varphi)\right]\), the planet fringes assume the same gray value along the line \(\varphi=\varphi_{P}\). The lines of equal gray value for the planet are aligned parallel to this direction. The fringes of the planet and the star are directed along different directions for a general planetary-system configuration. While the planet moves along its orbit around the star, it changes its azimuthal angle in time, \(\varphi_{p}(t)\). Therefore, the orientation of equal gray-value fringes parallel to the line \(\varphi=\varphi_{p}\) slowly changes. This may be discernable during long observations for the short-period Super-Earth type planets. It is likely undetectable for an Earth-like planet, due to its relatively long orbital period (365 Earth days). The trigonometric identity for the cosine of a double angle is applied to Equation (16). \[M_{SP}(\rho,\varphi;\varphi_{S},\theta_{S};\varphi_{P},\theta_{P})=\left\{ \begin{array}{l}+M_{S}\Big{\{}\cos^{2}[kp\ sin\theta_{S}\ sin(\Delta\varphi)\ \sin(\varphi-\varphi_{S})]\Big{\}}\\ +M_{P}\Big{\{}\cos^{2}[kp\ sin\theta_{P}\ sin(\Delta\varphi)\ \sin(\varphi- \varphi_{P})]\Big{\}}\end{array} \right.