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 CartesianCoordinates by | (4) |
 | (5) |
 | |  | (6) |
|  | (7) | |
|  | (8) | |
 | |  | (9) |
| | (10) | |
| | (11) | |
 | |  | (12) |
The Laplacian is
 | (13) |
 | |  | (14) |
 | |  | (15) |
 | |  | (16) |
Then the new Scale Factors are
 | |  | (17) |
 | |  | (18) |
 | | | (19) |
The Helmholtz Differential Equation is Separable.See also Cylindrical Coordinates, Helmholtz Differential Equation--Elliptic Cylindrical Coordinates
References
Arfken, G. ``Elliptic Cylindrical Coordinates ( Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 657, 1953., , 
).'' §2.7 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 95-97, 1970.
- Lorentz Group
- Lorentzian Distribution
- Lorentzian Function
- Lorentz Tensor
- Lorentz Transformation
- Lorenz System
- Lorraine Cross
- Lo Shu
- Lotka-Volterra Equations
- Low-Dimensional Topology
- Lower Bound
- Lower Denjoy Sum
- Lower Integral
- Lower Limit
- Lower Sum
- Lower-Trimmed Subsequence
- Lowest Terms Fraction
- Loxodrome
- Lozenge
- Lozenge Method
- Lozi Map
- Lp'-Balance Theorem
- L-Polyomino
- L-Series
- L-System