Let us consider two cylindrical coordinate system $(r, \theta)$ and $(r_k, \theta_k)$ linked by the relation: \begin{align} re^{i\theta} = R_k e^{i\phi_k} + r_k e^{i\theta_k}, \end{align} where $R_k$ is the distance between the two system and $\phi_k$ the angle. Using this equation, we can prove the following relations: \begin{align} r_k^2 &= (r\cos\theta - R_k\cos\phi_k)^2 + (r\sin\theta - R_k\sin\phi_k)^2, \\ \tan\theta_k &= \frac{r\sin\theta - R_k\sin\phi_k}{r\cos\theta - R_k\cos\phi_k}. \end{align} When $r\to\infty$, these two relations simplify into: \begin{align} r_k &\approx r - (R_k\cos\phi_k\cos\theta + R_k\sin\phi_k\sin\theta), \\ \theta_k &\approx \theta. \end{align}
My problem is to obtain the same kind of relations between two elliptic coordinate system, $(\xi, \eta)$ and $(\xi_k, \eta_k)$ with linear eccentricity $c$ and $c_k$ respectively. Both coordinate systems are linked via the relation: \begin{align} c\cosh(\xi \pm i\eta) = R_ke^{\pm i\phi_k} + c_ke^{\pm i\alpha_k}\cosh(\xi_k\pm i\eta_k), \end{align} where $\alpha_k$ is the angle between the k coordinate system and the horizon.
My goal is to express $\xi_k$ and $\eta_k$ as a function of $\xi$ and $\eta$, respectively, whenever $\xi\to\infty$. I have done some tests and clearly we have: \begin{equation} \xi_k \approx \xi - cte, \end{equation} but I have not found the constant term yet.
Usign the fact that: \begin{equation} r^2 = c^2 \left(\cosh^2\xi \cos^2\eta + \sinh^2\xi\sin^2\eta\right) \underset{\xi\to\infty}{\sim} \left(\frac{ce^\xi}{2}\right)^2, \end{equation} we have: \begin{equation} c_ke^{\xi_k} \underset{\xi\to\infty}{\sim} ce^\xi - 2(R_k\cos\phi_k\cos\theta + R_k\sin\phi_k\sin\theta). \end{equation}
Also, we have: \begin{equation} \eta_k = \eta - \alpha_k \end{equation}