The relationship between the ray paths

At the reflector point R, the ray must obey Snell's law. In other words, the angle of incidence must equal the angle of reflection. In terms of ray paths, ${\bf r}_0$ fixes the position of the reflection plane, and ${\bf r}_s$ and ${\bf r}_g$ account for the angles of incidence and reflection. Since ${\bf r}_s$ and ${\bf r}_g$ have equal lengths 1/v, Snell's law can be expressed in the following vectorial form:  

$${\bf r}_0 = \lambda ({\bf r}_s + {\bf r}_g) ,$$ (9)

where $\lambda$ is a coefficient of proportionality. The first two coordinates are the ray parameter components, which have the following relation:  

$${\bf p}_0 = \lambda ({\bf p}_s + {\bf p}_g) .$$ (10)

The third coordinate of the ray path vectors can be expressed as a function of the inclination angle and the vector length, 1/v, as follows:  

$$r_z = \frac{\cos \theta(p,t)}{v} ,$$ (11)

where $\theta(p,t)$ is the table of angles along a ray of parameter p and time t. Then, after simplification of the 1/v factors (the velocity is the same for all rays at the reflector point), the third equation of relation (9) becomes  

$$\cos \theta(p_0,t_0) = \lambda ( \cos \theta(p_s,t_s) + \cos \theta(p_g,t_g) ) .$$ (12)

Substituting equation (12) into relation (10), we can eliminate the proportionality factor $\lambda$ to obtain the following relation:  

$${\bf p}_0 \left( \cos \theta(p_s,t_s) + \cos \theta(p_g,t_g ) \right) = ({\bf p}_s + {\bf p}_g) \cos \theta(p_0,t_0) .$$ (13)