Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system
(
λ
,
μ
,
ν
)
that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces, the ellipsoidal coordinate system is not produced by rotating or projecting any two-dimensional orthogonal coordinate system.
The Cartesian coordinates
(
x
,
y
,
z
)
can be produced from the ellipsoidal coordinates
(
λ
,
μ
,
ν
)
by the equations
x
2
=
(
a
2
+
λ
)
(
a
2
+
μ
)
(
a
2
+
ν
)
(
a
2
−
b
2
)
(
a
2
−
c
2
)
y
2
=
(
b
2
+
λ
)
(
b
2
+
μ
)
(
b
2
+
ν
)
(
b
2
−
a
2
)
(
b
2
−
c
2
)
z
2
=
(
c
2
+
λ
)
(
c
2
+
μ
)
(
c
2
+
ν
)
(
c
2
−
b
2
)
(
c
2
−
a
2
)
where the following limits apply to the coordinates
−
λ
<
c
2
<
−
μ
<
b
2
<
−
ν
<
a
2
.
Consequently, surfaces of constant
λ
are ellipsoids
x
2
a
2
+
λ
+
y
2
b
2
+
λ
+
z
2
c
2
+
λ
=
1
,
whereas surfaces of constant
μ
are hyperboloids of one sheet
x
2
a
2
+
μ
+
y
2
b
2
+
μ
+
z
2
c
2
+
μ
=
1
,
because the last term in the lhs is negative, and surfaces of constant
ν
are hyperboloids of two sheets
x
2
a
2
+
ν
+
y
2
b
2
+
ν
+
z
2
c
2
+
ν
=
1
because the last two terms in the lhs are negative.
Scale factors and differential operators
For brevity in the equations below, we introduce a function
S
(
σ
)
=
d
e
f
(
a
2
+
σ
)
(
b
2
+
σ
)
(
c
2
+
σ
)
where
σ
can represent any of the three variables
(
λ
,
μ
,
ν
)
. Using this function, the scale factors can be written
h
λ
=
1
2
(
λ
−
μ
)
(
λ
−
ν
)
S
(
λ
)
h
μ
=
1
2
(
μ
−
λ
)
(
μ
−
ν
)
S
(
μ
)
h
ν
=
1
2
(
ν
−
λ
)
(
ν
−
μ
)
S
(
ν
)
Hence, the infinitesimal volume element equals
d
V
=
(
λ
−
μ
)
(
λ
−
ν
)
(
μ
−
ν
)
8
−
S
(
λ
)
S
(
μ
)
S
(
ν
)
d
λ
d
μ
d
ν
and the Laplacian is defined by
∇
2
Φ
=
4
S
(
λ
)
(
λ
−
μ
)
(
λ
−
ν
)
∂
∂
λ
[
S
(
λ
)
∂
Φ
∂
λ
]
+
Other differential operators such as
∇
⋅
F
and
∇
×
F
can be expressed in the coordinates
(
λ
,
μ
,
ν
)
by substituting the scale factors into the general formulae found in orthogonal coordinates.
