Classically, we have the relation between cartesian and cylindrical coordinates given by: \begin{align} x &= r\cos(\theta),\\ y &= r\sin(\theta), \end{align} such that we have: \begin{equation} x\cos(\phi_0)+y\sin(\phi_0) = r\cos(\theta-\phi_0). \end{equation}
The relation between cartesian and elliptic-cylindrical coordinates is given by: \begin{align} x &= c\cosh(u)\cos(v),\\ y &= c\sinh(u)\sin(v). \end{align} For simplicity, let us consider that $c=1$, such that we have: \begin{equation} x\cos(\phi_0)+y\sin(\phi_0) = \cosh(u)\cos(v)\cos(\phi_0) + \sinh(u)\sin(v)\sin(\phi_0) \end{equation}
Is there a way to simplify this last equation into something similar than with cylindrical coordinates ?
The best would be to have something like : $x\cos(\phi_0)+y\sin(\phi_0) = \cosh(u+p)\cos(v+q)$. For now all I can obtain is: \begin{align} x\cos(\phi_0)+y\sin(\phi_0) &= \cosh(u)\frac{\cos(v-\phi_0)+\cos(v+\phi_0)}{2} \\&+ \sinh(u)\frac{\cos(v-\phi_0)-\cos(v+\phi_0)}{2} \end{align}