Elliptic Cylindrical Coordinates

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.

$$x = a\cosh u\cos v$$ (1)

$$y = a\sinh u\sin v$$ (2)

$$z = z,$$ (3)

where $u \in [0, \infty)$, $v \in [0, 2\pi)$, and $z \in (-\infty,\infty)$. They are related to Cartesian Coordinates by

$${x^2\over a^2\cosh^2 u}+{y^2\over a^2\sinh^2 u}=1$$ (4)

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

The Scale Factors are

$$h_1 = a\sqrt{\cosh^2 u\sin^2 v+\sinh^2 u\cos^2 v}$$ (6)

$$= a\sqrt{\cosh(2u)-\cos(2v)\over 2}$$ (7)

$$= a\sqrt{\sinh^2 u+\sin^2 v}$$ (8)

$$h_2 = a\sqrt{\sinh^2 u\sin^2 v+\sinh^2 u\cos^2 v}$$ (9)

$$= a\sqrt{\cosh(2u)-\cos(2v)\over 2}$$ (10)

$$= a\sqrt{\sinh^2 u+\sin^2 v}$$ (11)

$$h_3 = 1.$$ (12)

The Laplacian is

$$\nabla^2 = {1\over a^2(\sinh^2 u+\sin^2 v)}\left({{\partial^... ...l^2\over \partial v^2}}\right)+{\partial^2\over \partial z^2}.$$ (13)

Let

$$q_1 = \cosh u$$ (14)

$$q_2 = \cos v$$ (15)

$$q_3 = z.$$ (16)

Then the new Scale Factors are

$$h_{q_1} = a\sqrt{{q_1}^2-{q_2}^2\over {q_1}^2-1}$$ (17)

$$h_{q_2} = a\sqrt{{q_1}^2-{q_2}^2\over 1-{q_1}^2}$$ (18)

$$h_{q_3} = 1.$$ (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.