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Elliptic Cylindrical Coordinates

\begin{figure}\begin{center}\BoxedEPSF{curv_coords_Ellipt_Cyl.epsf}\end{center}\end{figure}

The $v$ coordinates are the asymptotic angle of confocal Parabola segments symmetrical about the $x$ axis. The $u$ coordinates are confocal Ellipses centered on the origin.

$\displaystyle x$ $\textstyle =$ $\displaystyle a\cosh u\cos v$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle a\sinh u\sin v$ (2)
$\displaystyle z$ $\textstyle =$ $\displaystyle z,$ (3)

where $u \in [0, \infty)$, $v \in [0, 2\pi)$, and $z \in (-\infty,\infty)$. They are related to Cartesian Coordinates by
\begin{displaymath}
{x^2\over a^2\cosh^2 u}+{y^2\over a^2\sinh^2 u}=1
\end{displaymath} (4)


\begin{displaymath}
{x^2\over a^2\cos^2 v}-{y^2\over a^2\sin^2 v}=1.
\end{displaymath} (5)

The Scale Factors are
$\displaystyle h_1$ $\textstyle =$ $\displaystyle a\sqrt{\cosh^2 u\sin^2 v+\sinh^2 u\cos^2 v}$ (6)
  $\textstyle =$ $\displaystyle a\sqrt{\cosh(2u)-\cos(2v)\over 2}$ (7)
  $\textstyle =$ $\displaystyle a\sqrt{\sinh^2 u+\sin^2 v}$ (8)
$\displaystyle h_2$ $\textstyle =$ $\displaystyle a\sqrt{\sinh^2 u\sin^2 v+\sinh^2 u\cos^2 v}$ (9)
  $\textstyle =$ $\displaystyle a\sqrt{\cosh(2u)-\cos(2v)\over 2}$ (10)
  $\textstyle =$ $\displaystyle a\sqrt{\sinh^2 u+\sin^2 v}$ (11)
$\displaystyle h_3$ $\textstyle =$ $\displaystyle 1.$ (12)

The Laplacian is
\begin{displaymath}
\nabla^2 = {1\over a^2(\sinh^2 u+\sin^2 v)}\left({{\partial^...
...l^2\over \partial v^2}}\right)+{\partial^2\over \partial z^2}.
\end{displaymath} (13)

Let
$\displaystyle q_1$ $\textstyle =$ $\displaystyle \cosh u$ (14)
$\displaystyle q_2$ $\textstyle =$ $\displaystyle \cos v$ (15)
$\displaystyle q_3$ $\textstyle =$ $\displaystyle z.$ (16)

Then the new Scale Factors are
$\displaystyle h_{q_1}$ $\textstyle =$ $\displaystyle a\sqrt{{q_1}^2-{q_2}^2\over {q_1}^2-1}$ (17)
$\displaystyle h_{q_2}$ $\textstyle =$ $\displaystyle a\sqrt{{q_1}^2-{q_2}^2\over 1-{q_1}^2}$ (18)
$\displaystyle h_{q_3}$ $\textstyle =$ $\displaystyle 1.$ (19)

The Helmholtz Differential Equation is Separable.

See also Cylindrical Coordinates, Helmholtz Differential Equation--Elliptic Cylindrical Coordinates


References

Arfken, G. ``Elliptic Cylindrical Coordinates ($u$, $v$, $z$).'' §2.7 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 95-97, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 657, 1953.



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© 1996-9 Eric W. Weisstein
1999-05-25