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

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

On the complex plane, an equivalent relationship is

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

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

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