## Elliptic Cylindrical Coordinates

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

 (1) (2) (3)

where , , and . They are related to Cartesian Coordinates by
 (4)

 (5)

The Scale Factors are
 (6) (7) (8) (9) (10) (11) (12)

The Laplacian is
 (13)

Let
 (14) (15) (16)

Then the new Scale Factors are
 (17) (18) (19)

The Helmholtz Differential Equation is Separable.

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