Elliptic coordinate system

In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae.

-axis of the Cartesian coordinate system.

The most common definition of elliptic coordinates

is a nonnegative real number and

On the complex plane, an equivalent relationship is These definitions correspond to ellipses and hyperbolae.

The trigonometric identity shows that curves of constant

form ellipses, whereas the hyperbolic trigonometric identity shows that curves of constant

In an orthogonal coordinate system the lengths of the basis vectors are known as scale factors.

The scale factors for the elliptic coordinates

are equal to Using the double argument identities for hyperbolic functions and trigonometric functions, the scale factors can be equivalently expressed as Consequently, an infinitesimal element of area equals and the Laplacian reads Other differential operators such as

by substituting the scale factors into the general formulae found in orthogonal coordinates.

An alternative and geometrically intuitive set of elliptic coordinates

σ = cosh ⁡ μ

τ = cos ⁡ ν

are ellipses, whereas the curves of constant

have a simple relation to the distances to the foci

For any point in the plane, the sum

of its distances to the foci equals

, so the conversion to Cartesian coordinates is not a function, but a multifunction.

The scale factors for the alternative elliptic coordinates

are Hence, the infinitesimal area element becomes and the Laplacian equals Other differential operators such as

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: 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

that sum to a fixed vector

, where the integrand was a function of the vector lengths

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).