Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length.

Prolate spheroidal coordinates can be used to solve various partial differential equations in which the boundary conditions match its symmetry and shape, such as solving for a field produced by two centers, which are taken as the foci on the z-axis. One example is solving for the wavefunction of an electron moving in the electromagnetic field of two positively charged nuclei, as in the hydrogen molecular ion, H₂⁺. Another example is solving for the electric field generated by two small electrode tips. Other limiting cases include areas generated by a line segment (μ = 0) or a line with a missing segment (ν=0). The electronic structure of general diatomic molecules with many electrons can also be solved to excellent precision in the prolate spheroidal coordinate system.^[1]

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

${\displaystyle x=a\sinh \mu \sin \nu \cos \varphi }$

${\displaystyle y=a\sinh \mu \sin \nu \sin \varphi }$

${\displaystyle z=a\cosh \mu \cos \nu }$

where ${\displaystyle \mu }$ is a nonnegative real number and ${\displaystyle \nu \in [0,\pi ]}$. The azimuthal angle ${\displaystyle \varphi }$ belongs to the interval ${\displaystyle [0,2\pi ]}$.

The trigonometric identity

${\displaystyle {\frac {z^{2}}{a^{2}\cosh ^{2}\mu }}+{\frac {x^{2}+y^{2}}{a^{2}\sinh ^{2}\mu }}=\cos ^{2}\nu +\sin ^{2}\nu =1}$

shows that surfaces of constant ${\displaystyle \mu }$ form prolate spheroids, since they are ellipses rotated about the axis joining their foci. Similarly, the hyperbolic trigonometric identity

${\displaystyle {\frac {z^{2}}{a^{2}\cos ^{2}\nu }}-{\frac {x^{2}+y^{2}}{a^{2}\sin ^{2}\nu }}=\cosh ^{2}\mu -\sinh ^{2}\mu =1}$

shows that surfaces of constant ${\displaystyle \nu }$ form hyperboloids of revolution.

The distances from the foci located at ${\displaystyle (x,y,z)=(0,0,\pm a)}$ are

${\displaystyle r_{\pm }={\sqrt {x^{2}+y^{2}+(z\mp a)^{2}}}=a(\cosh \mu \mp \cos \nu ).}$

The scale factors for the elliptic coordinates ${\displaystyle (\mu ,\nu )}$ are equal

${\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {\sinh ^{2}\mu +\sin ^{2}\nu }}}$

whereas the azimuthal scale factor is

${\displaystyle h_{\varphi }=a\sinh \mu \sin \nu ,}$

resulting in a metric of

${\displaystyle {\begin{aligned}ds^{2}&=h_{\mu }^{2}d\mu ^{2}+h_{\nu }^{2}d\nu ^{2}+h_{\varphi }^{2}d\varphi ^{2}\\&=a^{2}\left[(\sinh ^{2}\mu +\sin ^{2}\nu )d\mu ^{2}+(\sinh ^{2}\mu +\sin ^{2}\nu )d\nu ^{2}+(\sinh ^{2}\mu \sin ^{2}\nu )d\varphi ^{2}\right].\end{aligned}}}$

Consequently, an infinitesimal volume element equals

${\displaystyle dV=a^{3}\sinh \mu \sin \nu (\sinh ^{2}\mu +\sin ^{2}\nu )\,d\mu \,d\nu \,d\varphi }$

and the Laplacian can be written

${\displaystyle {\begin{aligned}\nabla ^{2}\Phi ={}&{\frac {1}{a^{2}(\sinh ^{2}\mu +\sin ^{2}\nu )}}\left[{\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}+\coth \mu {\frac {\partial \Phi }{\partial \mu }}+\cot \nu {\frac {\partial \Phi }{\partial \nu }}\right]\\[6pt]&{}+{\frac {1}{a^{2}\sinh ^{2}\mu \sin ^{2}\nu }}{\frac {\partial ^{2}\Phi }{\partial \varphi ^{2}}}\end{aligned}}}$

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

An alternative and geometrically intuitive set of prolate spheroidal coordinates ${\displaystyle (\sigma ,\tau ,\phi )}$ are sometimes used, where ${\displaystyle \sigma =\cosh \mu }$ and ${\displaystyle \tau =\cos \nu }$. Hence, the curves of constant ${\displaystyle \sigma }$ are prolate spheroids, whereas the curves of constant ${\displaystyle \tau }$ are hyperboloids of revolution. The coordinate ${\displaystyle \tau }$ belongs to the interval [−1, 1], whereas the ${\displaystyle \sigma }$ coordinate must be greater than or equal to one.

The coordinates ${\displaystyle \sigma }$ and ${\displaystyle \tau }$ have a simple relation to the distances to the foci ${\displaystyle F_{1}}$ and ${\displaystyle F_{2}}$. For any point in the plane, the sum ${\displaystyle d_{1}+d_{2}}$ of its distances to the foci equals ${\displaystyle 2a\sigma }$, whereas their difference ${\displaystyle d_{1}-d_{2}}$ 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 ${\displaystyle z=-a}$ and ${\displaystyle z=+a}$, respectively.) This gives the following expressions for ${\displaystyle \sigma }$, ${\displaystyle \tau }$, and ${\displaystyle \varphi }$:

${\displaystyle \sigma ={\frac {1}{2a}}\left({\sqrt {x^{2}+y^{2}+(z+a)^{2}}}+{\sqrt {x^{2}+y^{2}+(z-a)^{2}}}\right)}$

${\displaystyle \tau ={\frac {1}{2a}}\left({\sqrt {x^{2}+y^{2}+(z+a)^{2}}}-{\sqrt {x^{2}+y^{2}+(z-a)^{2}}}\right)}$

${\displaystyle \varphi =\arctan \left({\frac {y}{x}}\right)}$

Unlike the analogous oblate spheroidal coordinates, the prolate spheroid coordinates (σ, τ, φ) are not degenerate; in other words, there is a unique, reversible correspondence between them and the Cartesian coordinates

${\displaystyle x=a{\sqrt {(\sigma ^{2}-1)(1-\tau ^{2})}}\cos \varphi }$

${\displaystyle y=a{\sqrt {(\sigma ^{2}-1)(1-\tau ^{2})}}\sin \varphi }$

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

The scale factors for the alternative elliptic coordinates ${\displaystyle (\sigma ,\tau ,\varphi )}$ are

${\displaystyle h_{\sigma }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{\sigma ^{2}-1}}}}$

${\displaystyle h_{\tau }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{1-\tau ^{2}}}}}$

while the azimuthal scale factor is now

${\displaystyle h_{\varphi }=a{\sqrt {\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}}}$

Hence, the infinitesimal volume element becomes

${\displaystyle dV=a^{3}(\sigma ^{2}-\tau ^{2})\,d\sigma \,d\tau \,d\varphi }$

and the Laplacian equals

${\displaystyle {\begin{aligned}\nabla ^{2}\Phi ={}&{\frac {1}{a^{2}(\sigma ^{2}-\tau ^{2})}}\left\{{\frac {\partial }{\partial \sigma }}\left[\left(\sigma ^{2}-1\right){\frac {\partial \Phi }{\partial \sigma }}\right]+{\frac {\partial }{\partial \tau }}\left[(1-\tau ^{2}){\frac {\partial \Phi }{\partial \tau }}\right]\right\}\\&{}+{\frac {1}{a^{2}(\sigma ^{2}-1)(1-\tau ^{2})}}{\frac {\partial ^{2}\Phi }{\partial \varphi ^{2}}}\end{aligned}}}$

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

As is the case with spherical coordinates, Laplace's equation may be solved by the method of separation of variables to yield solutions in the form of prolate spheroidal harmonics, which are convenient to use when boundary conditions are defined on a surface with a constant prolate spheroidal coordinate (See Smythe, 1968).