Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular ${\displaystyle z}$-direction. Hence, the coordinate surfaces are prisms of 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 cylindrical coordinates ${\displaystyle (\mu ,\nu ,z)}$ is

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

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.

The scale factors for the elliptic cylindrical coordinates ${\displaystyle \mu }$ and ${\displaystyle \nu }$ are equal

whereas the remaining scale factor ${\displaystyle h_{z}=1}$. Consequently, an infinitesimal volume element equals