In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci 👁 {\displaystyle F_{1}}
and 👁 {\displaystyle F_{2}}
are generally taken to be fixed at 👁 {\displaystyle -a}
and 👁 {\displaystyle +a}
, respectively, on the 👁 {\displaystyle x}
-axis of the Cartesian coordinate system.

The most common definition of elliptic coordinates 👁 {\displaystyle (\mu ,\nu )}
is

👁 {\displaystyle {\begin{aligned}x&=a\ \cosh \mu \ \cos \nu \\y&=a\ \sinh \mu \ \sin \nu \end{aligned}}}

where 👁 {\displaystyle \mu }
is a nonnegative real number and 👁 {\displaystyle \nu \in [0,2\pi ].}

On the complex plane, an equivalent relationship is

👁 {\displaystyle x+iy=a\ \cosh(\mu +i\nu )}

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

👁 {\displaystyle {\frac {x^{2}}{a^{2}\cosh ^{2}\mu }}+{\frac {y^{2}}{a^{2}\sinh ^{2}\mu }}=\cos ^{2}\nu +\sin ^{2}\nu =1}

shows that curves of constant 👁 {\displaystyle \mu }
form ellipses, whereas the hyperbolic trigonometric identity

👁 {\displaystyle {\frac {x^{2}}{a^{2}\cos ^{2}\nu }}-{\frac {y^{2}}{a^{2}\sin ^{2}\nu }}=\cosh ^{2}\mu -\sinh ^{2}\mu =1}

shows that curves of constant 👁 {\displaystyle \nu }
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 👁 {\displaystyle (\mu ,\nu )}
are equal to

👁 {\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {\sinh ^{2}\mu +\sin ^{2}\nu }}=a{\sqrt {\cosh ^{2}\mu -\cos ^{2}\nu }}.}

Using the double argument identities for hyperbolic functions and trigonometric functions, the scale factors can be equivalently expressed as

👁 {\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {{\frac {1}{2}}(\cosh 2\mu -\cos 2\nu )}}.}

Consequently, an infinitesimal element of area equals

👁 {\displaystyle {\begin{aligned}dA&=h_{\mu }h_{\nu }d\mu d\nu \\&=a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)d\mu d\nu \\&=a^{2}\left(\cosh ^{2}\mu -\cos ^{2}\nu \right)d\mu d\nu \\&={\frac {a^{2}}{2}}\left(\cosh 2\mu -\cos 2\nu \right)d\mu d\nu \end{aligned}}}

and the Laplacian reads

👁 {\displaystyle {\begin{aligned}\nabla ^{2}\Phi &={\frac {1}{a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)\\&={\frac {1}{a^{2}\left(\cosh ^{2}\mu -\cos ^{2}\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)\\&={\frac {2}{a^{2}\left(\cosh 2\mu -\cos 2\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)\end{aligned}}}

Other differential operators such as 👁 {\displaystyle \nabla \cdot \mathbf {F} }
and 👁 {\displaystyle \nabla \times \mathbf {F} }
can be expressed in the coordinates 👁 {\displaystyle (\mu ,\nu )}
by substituting the scale factors into the general formulae found in orthogonal coordinates.

An alternative and geometrically intuitive set of elliptic coordinates 👁 {\displaystyle (\sigma ,\tau )}
are sometimes used, where 👁 {\displaystyle \sigma =\cosh \mu }
and 👁 {\displaystyle \tau =\cos \nu }
. Hence, the curves of constant 👁 {\displaystyle \sigma }
are ellipses, whereas the curves of constant 👁 {\displaystyle \tau }
are hyperbolae. The coordinate 👁 {\displaystyle \tau }
must belong to the interval [-1, 1], whereas the 👁 {\displaystyle \sigma }
coordinate must be greater than or equal to one.

The coordinates 👁 {\displaystyle (\sigma ,\tau )}
have a simple relation to the distances to the foci 👁 {\displaystyle F_{1}}
and 👁 {\displaystyle F_{2}}
. For any point in the plane, the sum 👁 {\displaystyle d_{1}+d_{2}}
of its distances to the foci equals 👁 {\displaystyle 2a\sigma }
, whereas their difference 👁 {\displaystyle d_{1}-d_{2}}
equals 👁 {\displaystyle 2a\tau }
. Thus, the distance to 👁 {\displaystyle F_{1}}
is 👁 {\displaystyle a(\sigma +\tau )}
, whereas the distance to 👁 {\displaystyle F_{2}}
is 👁 {\displaystyle a(\sigma -\tau )}
. (Recall that 👁 {\displaystyle F_{1}}
and 👁 {\displaystyle F_{2}}
are located at 👁 {\displaystyle x=-a}
and 👁 {\displaystyle x=+a}
, respectively.)

A drawback of these coordinates is that the points with Cartesian coordinates (x,y) and (x,-y) have the same coordinates 👁 {\displaystyle (\sigma ,\tau )}
, so the conversion to Cartesian coordinates is not a function, but a multifunction.

👁 {\displaystyle x=a\left.\sigma \right.\tau }

👁 {\displaystyle y^{2}=a^{2}\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right).}

The scale factors for the alternative elliptic coordinates 👁 {\displaystyle (\sigma ,\tau )}
are

👁 {\displaystyle h_{\sigma }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{\sigma ^{2}-1}}}}

👁 {\displaystyle h_{\tau }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{1-\tau ^{2}}}}.}

Hence, the infinitesimal area element becomes

👁 {\displaystyle dA=a^{2}{\frac {\sigma ^{2}-\tau ^{2}}{\sqrt {\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}}}d\sigma d\tau }

and the Laplacian equals

👁 {\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sigma ^{2}-\tau ^{2}\right)}}\left[{\sqrt {\sigma ^{2}-1}}{\frac {\partial }{\partial \sigma }}\left({\sqrt {\sigma ^{2}-1}}{\frac {\partial \Phi }{\partial \sigma }}\right)+{\sqrt {1-\tau ^{2}}}{\frac {\partial }{\partial \tau }}\left({\sqrt {1-\tau ^{2}}}{\frac {\partial \Phi }{\partial \tau }}\right)\right].}

Other differential operators such as 👁 {\displaystyle \nabla \cdot \mathbf {F} }
and 👁 {\displaystyle \nabla \times \mathbf {F} }
can be expressed in the coordinates 👁 {\displaystyle (\sigma ,\tau )}
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:

1. The elliptic cylindrical coordinates are produced by projecting in the 👁 {\displaystyle z}
   -direction.
2. The prolate spheroidal coordinates are produced by rotating the elliptic coordinates about the 👁 {\displaystyle x}
   -axis, i.e., the axis connecting the foci, whereas the oblate spheroidal coordinates are produced by rotating the elliptic coordinates about the 👁 {\displaystyle y}
   -axis, i.e., the axis separating the foci.
3. Ellipsoidal coordinates are a formal extension of elliptic coordinates into 3-dimensions, which is based on confocal ellipsoids, hyperboloids of one and two sheets.

Note that (ellipsoidal) Geographic coordinate system is a different concept from above.

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 👁 {\displaystyle \mathbf {p} }
and 👁 {\displaystyle \mathbf {q} }
that sum to a fixed vector 👁 {\displaystyle \mathbf {r} =\mathbf {p} +\mathbf {q} }
, where the integrand was a function of the vector lengths 👁 {\displaystyle \left|\mathbf {p} \right|}
and 👁 {\displaystyle \left|\mathbf {q} \right|}
. (In such a case, one would position 👁 {\displaystyle \mathbf {r} }
between the two foci and aligned with the 👁 {\displaystyle x}
-axis, i.e., 👁 {\displaystyle \mathbf {r} =2a\mathbf {\hat {x}} }
.) For concreteness, 👁 {\displaystyle \mathbf {r} }
, 👁 {\displaystyle \mathbf {p} }
and 👁 {\displaystyle \mathbf {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).

• Curvilinear coordinates
• Ellipsoidal coordinates
• Generalized coordinates
• Bipolar coordinates

• "Elliptic coordinates", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill.
• Weisstein, Eric W. "Elliptic Cylindrical Coordinates." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/EllipticCylindricalCoordinates.html