I'm going to find a relation between cartesian and elliptic unit vectors.
For ellipse with major a and minor semiaxes $a$ and $b$ cartesian coordinates are:
$x=f \text{cosh}(\xi)\text{cos}(\eta)$;
$y=f \text{sinh}(\xi)\text{sin}(\eta)$,
where $f = \sqrt{a^2-b^2}$.
Elliptic coordinates are:
$\xi = Re(\text{acosh}( {{x + iy} \over {f}} )) $;
$\eta = Im(\text{acosh}( {{x + iy} \over {f}} )) $,
(this equations are taken from [1, 9.11 p.170]).
According to [2, (6) p.481] elliptic unit vectors are related to cartesian by this matrix equation:
${\begin{pmatrix} \mathbf{\vec{e_x}} \\ \mathbf{\vec{e_y}} \end{pmatrix}} = { \frac{1}{\sqrt{D}} } {\begin{pmatrix}
\text{sinh}(\xi)\text{cos}(\eta) & -\text{cosh}(\xi)\text{sin}(\eta) \\
\text{cosh}(\xi)\text{sin}(\eta) & \text{sinh}(\xi)\text{sin}(\eta)
\end{pmatrix}} {\begin{pmatrix} \mathbf{\vec{e_\xi}} \\ \mathbf{\vec{e_\eta}} \end{pmatrix}} $,
where $ D = { 1 \over 2 } (\text{cosh}(2\xi) - \text{cos}(2\eta)) $ is a determinant of a matrix (see above).
This matrix is build from a partial derivatives
$ \begin{pmatrix}
\frac{\partial x}{\partial \xi} & \frac{\partial x}{\partial \eta}
\\
\frac{\partial y}{\partial \xi} & \frac{\partial y}{\partial \eta}
\end{pmatrix} $ (AFAIU, Jacobian, see [3])..
I need to find the matrix $M$ of inverse transformation:
${\begin{pmatrix} \mathbf{\vec{e_\xi}} \\ \mathbf{\vec{e_\eta}} \end{pmatrix}} = M {\begin{pmatrix} \mathbf{\vec{e_x}} \\ \mathbf{\vec{e_y}} \end{pmatrix}} $.
Any ideas?
References:
1. J. C. Gutiérrez-Vega and S. Chávez-Cerda, "Probability distributions in classical and quantum elliptic billiards," Revista Mexicana de Física, Vol. 47, no. 5, pp. 480-488, Oct. 2001.
2. N. W. McLachlan, "Theory and applications of Mathieu functions," Oxford University Press, 1937. 3. http://en.wikipedia.org/wiki/Curvilinear_coordinates
