Constant Coordinate Curves for Elliptic Coordinates

​
interfocal separation
This Demonstration shows curves of constant coordinate values for the elliptic coordinate system in two dimensions. These curves are semi-hyperbolas and ellipses, the latter having foci at
(±a,0)
. As you drag the locator in the
xy
plane, the curves are redrawn so they pass through that point. Holding the mouse over the curve shows which variable is constant along that curve, and holding it over the point gives the actual values of the variables. You can vary the interfocal separation,
2a
; at
a=0
the elliptic coordinates are equivalent to polar coordinates.

Details

Two-dimensional elliptic coordinates
(ξ,η)
may be defined by
x=acosh(ξ)cos(η)
,
y=asinh(ξ)sin(η)
, for
ξ∈[0,∞)
and
η∈(-π,π]
, with
a>0
representing the interfocal separation. The curves of constant
ξ
and
η
are ellipses and hyperbolas, respectively.
The inverse relation can be expressed
ξ=
-1
cosh
r
A
+
r
B
2a
,
η=(2θ(y)-1)
-1
cos
r
A
-
r
B
2a
, where
r
A,B
=
2
(x±a)
+
2
y
is the distance from the left/right focus. The slightly ungainly factor
2θ(y)-1
is necessary here to ensure that the correct (upper/lower) half-plane is chosen.
In the limit
a0
, the elliptic coordinates reduce to polar coordinates
(r,θ)
. The correspondence is given by
ηθ
and
acosh(ξ)r
(note that
ξ
itself becomes infinite as
a0
).
Three-dimensional generalizations of the elliptic coordinates are the oblate and prolate spheroidal coordinates, elliptic cylindrical coordinates, and ellipsoidal coordinates.

External Links

Elliptic Cylindrical Coordinates (Wolfram MathWorld)
Confocal Ellipsoidal Coordinates (Wolfram MathWorld)
Prolate Spheroidal Coordinates (Wolfram MathWorld)

Permanent Citation

Peter Falloon
​
​"Constant Coordinate Curves for Elliptic Coordinates"​
​http://demonstrations.wolfram.com/ConstantCoordinateCurvesForEllipticCoordinates/​
​Wolfram Demonstrations Project​
​Published: April 16, 2025