Prolate spheroidal coordinates

Prolate spheroidal coordinates μ and ν for a = 1. The lines of equal values of μ and ν are shown on the xz-plane, i.e. for φ = 0. The surfaces of constant μ and ν are obtained by rotation about the z-axis, so that the diagram is valid for any plane containing the z-axis: i.e. for any φ.

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.

In principle, a definition of prolate spheroidal coordinates could be degenerate. In other words, a single set of coordinates might correspond to two points in Cartesian coordinates; this is illustrated here with two black spheres, one on each sheet of the hyperboloid and located at (x, y, ±z). However, neither of the definitions presented here are degenerate.

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).

• MathWorld description of prolate spheroidal coordinates