Solving the system

The second relation of system (14) can be isolated because it simply adds two relations and two unknowns (the coordinates of ${\bf x}_0$). Similarly, the third equation can be substituted into the others by replacing t_s (or t_g) with t_sg - t_g (or t_sg - t_s).

Thus, we obtain a reduced system of four relations (six equations):  

$$\left\{ \begin{array} {ccc} \xi(p_g, t_g)\frac{{\bf p}_g}{... ... p}_s + {\bf p}_g) \cos \theta(p_0,t_0) \end{array} \right. .$$ (15)

and two additional relations to compute the remaining unknowns:  

$${\bf x}_0 = \xi(p_0,t_0)\frac{{\bf p}_0}{p_0} - \frac{1}{2}... ...c{{\bf p}_s}{p_s} + \xi(p_g,t_g)\frac{{\bf p}_g}{p_g} \right)$$ (16)

 

t_s = t_sg - t_g . (17)

The six unknowns of system (15) are t₀, t_g (or t_s), ${\bf p}_s$ [including the two unknowns (p_xs,p_ys) or $(p_s,\phi_s)$], and ${\bf p}_g$ [(p_xg,p_yg) or $(p_g,\phi_g)$]. The parameters are t_sg, ${\bf p}_0$[including the two parameters (p_x0,p_y0) or $(p_0,\phi_0)$], ${\bf x}_s$, and ${\bf x}_g$. This system, like its 2-D counterpart, can be solved using the Newton-Raphson algorithm Press et al. (1986).