Elliptic coordinate system

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 ${\displaystyle F_{1}}${\displaystyle F_{1}} and ${\displaystyle F_{2}}${\displaystyle F_{2}} are generally taken to be fixed at ${\displaystyle -a}${\displaystyle -a} and ${\displaystyle +a}${\displaystyle +a}, respectively, on the ${\displaystyle x}${\displaystyle x}-axis of the Cartesian coordinate system.

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

${\displaystyle {\begin{aligned}x&=a\ \cosh \mu \ \cos \nu \\y&=a\ \sinh \mu \ \sin \nu \end{aligned}}}${\displaystyle {\begin{aligned}x&=a\ \cosh \mu \ \cos \nu \\y&=a\ \sinh \mu \ \sin \nu \end{aligned}}}

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

On the complex plane, an equivalent relationship is

${\displaystyle x+iy=a\ \cosh(\mu +i\nu )}${\displaystyle x+iy=a\ \cosh(\mu +i\nu )}

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

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

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

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

shows that curves of constant ${\displaystyle \nu }${\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 ${\displaystyle (\mu ,\nu )}${\displaystyle (\mu ,\nu )} are equal to

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

Using the double argument identities for hyperbolic functions and trigonometric functions, the scale factors can be equivalently expressed as

${\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {{\frac {1}{2}}(\cosh 2\mu -\cos 2\nu )}}.}${\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {{\frac {1}{2}}(\cosh 2\mu -\cos 2\nu )}}.}

Consequently, an infinitesimal element of area equals

${\displaystyle {\begin{aligned}dA&=h_{\mu }h_{\nu }d\mu d\nu \\&=a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)d\mu d\nu \\&=a^{2}\left(\cosh ^{2}\mu -\cos ^{2}\nu \right)d\mu d\nu \\&={\frac {a^{2}}{2}}\left(\cosh 2\mu -\cos 2\nu \right)d\mu d\nu \end{aligned}}}${\displaystyle {\begin{aligned}dA&=h_{\mu }h_{\nu }d\mu d\nu \\&=a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)d\mu d\nu \\&=a^{2}\left(\cosh ^{2}\mu -\cos ^{2}\nu \right)d\mu d\nu \\&={\frac {a^{2}}{2}}\left(\cosh 2\mu -\cos 2\nu \right)d\mu d\nu \end{aligned}}}

and the Laplacian reads

${\displaystyle {\begin{aligned}\nabla ^{2}\Phi &={\frac {1}{a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)\\&={\frac {1}{a^{2}\left(\cosh ^{2}\mu -\cos ^{2}\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)\\&={\frac {2}{a^{2}\left(\cosh 2\mu -\cos 2\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)\end{aligned}}}${\displaystyle {\begin{aligned}\nabla ^{2}\Phi &={\frac {1}{a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)\\&={\frac {1}{a^{2}\left(\cosh ^{2}\mu -\cos ^{2}\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)\\&={\frac {2}{a^{2}\left(\cosh 2\mu -\cos 2\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)\end{aligned}}}

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

An alternative and geometrically intuitive set of elliptic coordinates ${\displaystyle (\sigma ,\tau )}${\displaystyle (\sigma ,\tau )} are sometimes used, where ${\displaystyle \sigma =\cosh \mu }${\displaystyle \sigma =\cosh \mu } and ${\displaystyle \tau =\cos \nu }${\displaystyle \tau =\cos \nu }. Hence, the curves of constant ${\displaystyle \sigma }${\displaystyle \sigma } are ellipses, whereas the curves of constant ${\displaystyle \tau }${\displaystyle \tau } are hyperbolae. The coordinate ${\displaystyle \tau }${\displaystyle \tau } must belong to the interval [-1, 1], whereas the ${\displaystyle \sigma }${\displaystyle \sigma } coordinate must be greater than or equal to one.

The coordinates ${\displaystyle (\sigma ,\tau )}${\displaystyle (\sigma ,\tau )} have a simple relation to the distances to the foci ${\displaystyle F_{1}}${\displaystyle F_{1}} and ${\displaystyle F_{2}}${\displaystyle F_{2}}. For any point in the plane, the sum ${\displaystyle d_{1}+d_{2}}${\displaystyle d_{1}+d_{2}} of its distances to the foci equals ${\displaystyle 2a\sigma }${\displaystyle 2a\sigma }, whereas their difference ${\displaystyle d_{1}-d_{2}}${\displaystyle d_{1}-d_{2}} equals ${\displaystyle 2a\tau }${\displaystyle 2a\tau }. Thus, the distance to ${\displaystyle F_{1}}${\displaystyle F_{1}} is ${\displaystyle a(\sigma +\tau )}${\displaystyle a(\sigma +\tau )}, whereas the distance to ${\displaystyle F_{2}}${\displaystyle F_{2}} is ${\displaystyle a(\sigma -\tau )}${\displaystyle a(\sigma -\tau )}. (Recall that ${\displaystyle F_{1}}${\displaystyle F_{1}} and ${\displaystyle F_{2}}${\displaystyle F_{2}} are located at ${\displaystyle x=-a}${\displaystyle x=-a} and ${\displaystyle x=+a}${\displaystyle x=+a}, respectively.)

A drawback of these coordinates is that the points with Cartesian coordinates (x,y) and (x,-y) have the same coordinates ${\displaystyle (\sigma ,\tau )}${\displaystyle (\sigma ,\tau )}, so the conversion to Cartesian coordinates is not a function, but a multifunction.

${\displaystyle x=a\left.\sigma \right.\tau }${\displaystyle x=a\left.\sigma \right.\tau }

${\displaystyle y^{2}=a^{2}\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right).}${\displaystyle y^{2}=a^{2}\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right).}

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

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

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

Hence, the infinitesimal area element becomes

${\displaystyle dA=a^{2}{\frac {\sigma ^{2}-\tau ^{2}}{\sqrt {\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}}}d\sigma d\tau }${\displaystyle dA=a^{2}{\frac {\sigma ^{2}-\tau ^{2}}{\sqrt {\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}}}d\sigma d\tau }

and the Laplacian equals

${\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sigma ^{2}-\tau ^{2}\right)}}\left[{\sqrt {\sigma ^{2}-1}}{\frac {\partial }{\partial \sigma }}\left({\sqrt {\sigma ^{2}-1}}{\frac {\partial \Phi }{\partial \sigma }}\right)+{\sqrt {1-\tau ^{2}}}{\frac {\partial }{\partial \tau }}\left({\sqrt {1-\tau ^{2}}}{\frac {\partial \Phi }{\partial \tau }}\right)\right].}${\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sigma ^{2}-\tau ^{2}\right)}}\left[{\sqrt {\sigma ^{2}-1}}{\frac {\partial }{\partial \sigma }}\left({\sqrt {\sigma ^{2}-1}}{\frac {\partial \Phi }{\partial \sigma }}\right)+{\sqrt {1-\tau ^{2}}}{\frac {\partial }{\partial \tau }}\left({\sqrt {1-\tau ^{2}}}{\frac {\partial \Phi }{\partial \tau }}\right)\right].}

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

Elliptic coordinates form the basis for several sets of three-dimensional orthogonal coordinates:

1. The elliptic cylindrical coordinates are produced by projecting in the ${\displaystyle z}${\displaystyle z}-direction.
2. The prolate spheroidal coordinates are produced by rotating the elliptic coordinates about the ${\displaystyle x}${\displaystyle x}-axis, i.e., the axis connecting the foci, whereas the oblate spheroidal coordinates are produced by rotating the elliptic coordinates about the ${\displaystyle y}${\displaystyle y}-axis, i.e., the axis separating the foci.
3. Ellipsoidal coordinates are a formal extension of elliptic coordinates into 3-dimensions, which is based on confocal ellipsoids, hyperboloids of one and two sheets.

Note that (ellipsoidal) Geographic coordinate system is a different concept from above.

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 are a natural description of a system thus allowing a separation of variables in the partial differential equations. Some traditional examples are solving systems such as electrons orbiting a molecule or planetary orbits that have an elliptical shape.

The geometric properties of elliptic coordinates can also be useful. A typical example might involve an integration over all pairs of vectors ${\displaystyle \mathbf {p} }${\displaystyle \mathbf {p} } and ${\displaystyle \mathbf {q} }${\displaystyle \mathbf {q} } that sum to a fixed vector ${\displaystyle \mathbf {r} =\mathbf {p} +\mathbf {q} }${\displaystyle \mathbf {r} =\mathbf {p} +\mathbf {q} }, where the integrand was a function of the vector lengths ${\displaystyle \left|\mathbf {p} \right|}${\displaystyle \left|\mathbf {p} \right|} and ${\displaystyle \left|\mathbf {q} \right|}${\displaystyle \left|\mathbf {q} \right|}. (In such a case, one would position ${\displaystyle \mathbf {r} }${\displaystyle \mathbf {r} } between the two foci and aligned with the ${\displaystyle x}${\displaystyle x}-axis, i.e., ${\displaystyle \mathbf {r} =2a\mathbf {\hat {x}} }${\displaystyle \mathbf {r} =2a\mathbf {\hat {x}} }.) For concreteness, ${\displaystyle \mathbf {r} }${\displaystyle \mathbf {r} }, ${\displaystyle \mathbf {p} }${\displaystyle \mathbf {p} } and ${\displaystyle \mathbf {q} }${\displaystyle \mathbf {q} } 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).

• Curvilinear coordinates
• Ellipsoidal coordinates
• Generalized coordinates
• Bipolar coordinates

• "Elliptic coordinates", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
• Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill.
• Weisstein, Eric W. "Elliptic Cylindrical Coordinates." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/EllipticCylindricalCoordinates.html

• Two-dimensional coordinate systems
• Ellipses

• Articles with short description
• Short description is different from Wikidata