Ingeometry, theelliptic coordinate system is a two-dimensionalorthogonalcoordinate system in which thecoordinate lines areconfocal ellipses and hyperbolae. The twofoci${\displaystyle F_1}$ and${\displaystyle F_2}$ are generally taken to be fixed at${\displaystyle -a}$ and${\displaystyle +a}$, respectively, on the${\displaystyle x}$-axis of theCartesian coordinate system.

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

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

On thecomplex 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${\displaystyle \mu }$ formellipses, 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${\displaystyle \nu }$ formhyperbolae.

In anorthogonal 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 thedouble argument identities forhyperbolic functions andtrigonometric 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

and the Laplacian reads

Other differential operators such as${\displaystyle \mathrm{\nabla }\cdot \mathbf{F}}$ and${\displaystyle \mathrm{\nabla }\times \mathbf{F}}$ can be expressed in the coordinates${\displaystyle (\mu ,\nu )}$ by substituting the scale factors into the general formulae found inorthogonal 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, thesum${\displaystyle d_1+d_2}$ of its distances to the foci equals${\displaystyle 2a\sigma }$, whereas theirdifference${\displaystyle d_1-d_2}$ 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 withCartesian coordinates (x,y) and (x,-y) have the same coordinates${\displaystyle (\sigma ,\tau )}$, so the conversion to Cartesian coordinates is not a function, but amultifunction.

${\displaystyle x=a\sigma \tau }$

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

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

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

Other differential operators such as${\displaystyle \mathrm{\nabla }\cdot \mathbf{F}}$ and${\displaystyle \mathrm{\nabla }\times \mathbf{F}}$ can be expressed in the coordinates${\displaystyle (\sigma ,\tau )}$ by substituting the scale factors into the general formulae found inorthogonal coordinates.

Elliptic coordinates form the basis for several sets of three-dimensionalorthogonal coordinates:

1. Theelliptic cylindrical coordinates are produced by projecting in the${\displaystyle z}$-direction.
2. Theprolate spheroidal coordinates are produced by rotating the elliptic coordinates about the${\displaystyle x}$-axis, i.e., the axis connecting the foci, whereas theoblate 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 solvingpartial differential equations, e.g.,Laplace's equation or theHelmholtz equation, for which elliptic coordinates are a natural description of a system thus allowing aseparation of variables in thepartial 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.,${\displaystyle \mathbf{r}=2a\hat{\mathbf{x}}}$.) For concreteness,${\displaystyle \mathbf{r}}$,${\displaystyle \mathbf{p}}$ and${\displaystyle \mathbf{q}}$ could represent themomenta 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 andKorn 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

• Two-dimensional coordinate systems
• Ellipses

• Articles with short description
• Short description is different from Wikidata