Elliptic Cylindrical Coordinates

​
xacosh(u) cos(v), yasinh(u) sin(v), zz
u
z
v
a
axes
The elliptic cylindrical curvilinear coordinate system is one of the many coordinate systems that make the Laplace and Helmoltz differential equations separable. This system is used when simple boundary conditions on a segment in the
x
-
y
plane are specified, as in the computation of the electric field around an infinite rectangular conducting plate whose intersection with the
x
-
y
plane is the line segment with endpoints
(-a,0)
and
(a,0)
.

Details

The more opaque half of the hyperbolic cylinder has
u≥0
; the other half has
u<0
.
From the tutorial on Mathematica Vector Analysis Package: The elliptic cylindrical coordinate system
EllipticCylindrical[u,v,z,a]
, parameterized by
a
, is built around two foci separated by
2a
. Holding coordinate
u
constant while varying the other coordinates produces a family of confocal ellipses. Fixing coordinate
v
produces a family of confocal hyperbolas. The coordinate
z
specifies distance along the axis of common focus. The default value for parameter
a
is
1
.

External Links

Elliptic Cylindrical Coordinates (Wolfram MathWorld)
Laplace's Equation (Wolfram MathWorld)
Helmoltz Differential Equation (Wolfram MathWorld)
Constant Coordinate Curves for Elliptic Coordinates
Vector Analysis Package (Wolfram Documentation Center)
Parabolic Cylindrical Coordinates
Cylindrical Coordinates
Spherical Coordinates

Permanent Citation

Adriano Pascoletti
​
​"Elliptic Cylindrical Coordinates"​
​http://demonstrations.wolfram.com/EllipticCylindricalCoordinates/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011