In geometry, the elliptic(al) coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci
F
1
and
F
2
are generally taken to be fixed at
−
a
and
+
a
, respectively, on the
x
-axis of the Cartesian coordinate system.
The most common definition of elliptic coordinates
(
μ
,
ν
)
is
x
=
a
cosh
μ
cos
ν
y
=
a
sinh
μ
sin
ν
where
μ
is a nonnegative real number and
ν
∈
[
0
,
2
π
]
.
On the complex plane, an equivalent relationship is
x
+
i
y
=
a
cosh
(
μ
+
i
ν
)
These definitions correspond to ellipses and hyperbolae. The trigonometric identity
x
2
a
2
cosh
2
μ
+
y
2
a
2
sinh
2
μ
=
cos
2
ν
+
sin
2
ν
=
1
shows that curves of constant
μ
form ellipses, whereas the hyperbolic trigonometric identity
x
2
a
2
cos
2
ν
−
y
2
a
2
sin
2
ν
=
cosh
2
μ
−
sinh
2
μ
=
1
shows that curves of constant
ν
form hyperbolae.
In an orthogonal coordinate system the lengths of the basis vectors are known as scale factors. The scale factors for the elliptic coordinates
(
μ
,
ν
)
are equal to
h
μ
=
h
ν
=
a
sinh
2
μ
+
sin
2
ν
=
a
cosh
2
μ
−
cos
2
ν
.
Using the double argument identities for hyperbolic functions and trigonometric functions, the scale factors can be equivalently expressed as
h
μ
=
h
ν
=
a
1
2
(
cosh
2
μ
−
cos
2
ν
)
.
Consequently, an infinitesimal element of area equals
d
A
=
h
μ
h
ν
d
μ
d
ν
=
a
2
(
sinh
2
μ
+
sin
2
ν
)
d
μ
d
ν
=
a
2
(
cosh
2
μ
−
cos
2
ν
)
d
μ
d
ν
=
a
2
2
(
cosh
2
μ
−
cos
2
ν
)
d
μ
d
ν
and the Laplacian reads
∇
2
Φ
=
1
a
2
(
sinh
2
μ
+
sin
2
ν
)
(
∂
2
Φ
∂
μ
2
+
∂
2
Φ
∂
ν
2
)
=
1
a
2
(
cosh
2
μ
−
cos
2
ν
)
(
∂
2
Φ
∂
μ
2
+
∂
2
Φ
∂
ν
2
)
=
2
a
2
(
cosh
2
μ
−
cos
2
ν
)
(
∂
2
Φ
∂
μ
2
+
∂
2
Φ
∂
ν
2
)
.
Other differential operators such as
∇
⋅
F
and
∇
×
F
can be expressed in the coordinates
(
μ
,
ν
)
by substituting the scale factors into the general formulae found in orthogonal coordinates.
An alternative and geometrically intuitive set of elliptic coordinates
(
σ
,
τ
)
are sometimes used, where
σ
=
cosh
μ
and
τ
=
cos
ν
. Hence, the curves of constant
σ
are ellipses, whereas the curves of constant
τ
are hyperbolae. The coordinate
τ
must belong to the interval [-1, 1], whereas the
σ
coordinate must be greater than or equal to one.
The coordinates
(
σ
,
τ
)
have a simple relation to the distances to the foci
F
1
and
F
2
. For any point in the plane, the sum
d
1
+
d
2
of its distances to the foci equals
2
a
σ
, whereas their difference
d
1
−
d
2
equals
2
a
τ
. Thus, the distance to
F
1
is
a
(
σ
+
τ
)
, whereas the distance to
F
2
is
a
(
σ
−
τ
)
. (Recall that
F
1
and
F
2
are located at
x
=
−
a
and
x
=
+
a
, respectively.)
A drawback of these coordinates is that the points with Cartesian coordinates (x,y) and (x,-y) have the same coordinates
(
σ
,
τ
)
, so the conversion to Cartesian coordinates is not a function, but a multifunction.
x
=
a
σ
τ
y
2
=
a
2
(
σ
2
−
1
)
(
1
−
τ
2
)
.
The scale factors for the alternative elliptic coordinates
(
σ
,
τ
)
are
h
σ
=
a
σ
2
−
τ
2
σ
2
−
1
h
τ
=
a
σ
2
−
τ
2
1
−
τ
2
.
Hence, the infinitesimal area element becomes
d
A
=
a
2
σ
2
−
τ
2
(
σ
2
−
1
)
(
1
−
τ
2
)
d
σ
d
τ
and the Laplacian equals
∇
2
Φ
=
1
a
2
(
σ
2
−
τ
2
)
[
σ
2
−
1
∂
∂
σ
(
σ
2
−
1
∂
Φ
∂
σ
)
+
1
−
τ
2
∂
∂
τ
(
1
−
τ
2
∂
Φ
∂
τ
)
]
.
Other differential operators such as
∇
⋅
F
and
∇
×
F
can be expressed in the coordinates
(
σ
,
τ
)
by substituting the scale factors into the general formulae found in orthogonal coordinates.
Elliptic coordinates form the basis for several sets of three-dimensional orthogonal coordinates. The elliptic cylindrical coordinates are produced by projecting in the
z
-direction. The prolate spheroidal coordinates are produced by rotating the elliptic coordinates about the
x
-axis, i.e., the axis connecting the foci, whereas the oblate spheroidal coordinates are produced by rotating the elliptic coordinates about the
y
-axis, i.e., the axis separating the foci.
The classic applications of elliptic coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which elliptic coordinates are a natural description of a system thus allowing a separation of variables in the partial differential equations. Some traditional examples are solving systems such as electrons orbiting a molecule or planetary orbits that have an elliptical shape.
The geometric properties of elliptic coordinates can also be useful. A typical example might involve an integration over all pairs of vectors
p
and
q
that sum to a fixed vector
r
=
p
+
q
, where the integrand was a function of the vector lengths
|
p
|
and
|
q
|
. (In such a case, one would position
r
between the two foci and aligned with the
x
-axis, i.e.,
r
=
2
a
x
^
.) For concreteness,
r
,
p
and
q
could represent the momenta of a particle and its decomposition products, respectively, and the integrand might involve the kinetic energies of the products (which are proportional to the squared lengths of the momenta).