Elliptic coordinates are a twodimensional 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 xaxis of the Cartesian coordinate system.
Contents 
The most common definition of elliptic coordinates (μ,ν) is
where μ 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 μ form ellipses, whereas the hyperbolic trigonometric identity
shows that curves of constant ν form hyperbolae.
The scale factors for the elliptic coordinates (μ,ν) are equal
To simplify the computation of the scale factors, double angle identities can be used to express them equivalently as
Consequently, an infinitesimal element of area equals
and the Laplacian equals
Other differential operators such as and 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 = coshμ and τ = cosν. 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 2aσ, whereas their difference d_{1} − d_{2} equals 2aτ. 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 they do not have a 1to1 transformation to the Cartesian coordinates
The scale factors for the alternative elliptic coordinates (σ,τ) are
Hence, the infinitesimal area element becomes
and the Laplacian equals
Other differential operators such as and 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 threedimensional orthogonal coordinates. The elliptic cylindrical coordinates are produced by projecting in the zdirection. The prolate spheroidal coordinates are produced by rotating the elliptic coordinates about the xaxis, i.e., the axis connecting the foci, whereas the oblate spheroidal coordinates are produced by rotating the elliptic coordinates about the yaxis, 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 allow a separation of variables. A typical example would be the electric field surrounding a flat conducting plate of width 2a.
The geometric properties of elliptic coordinates can also be useful. A typical example might involve an integration over all pairs of vectors and that sum to a fixed vector , where the integrand was a function of the vector lengths and . (In such a case, one would position between the two foci and aligned with the xaxis, i.e., .) For concreteness, , and 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).
