Wikipedia / Elliptic coordinate system
Wikipedia / 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-axis of the Cartesian coordinate system.
Basic definition
is
On the complex plane, an equivalent relationship is
These definitions correspond to ellipses and hyperbolae. The trigonometric identity
form ellipses, whereas the hyperbolic trigonometric identity
form hyperbolae.
Scale factors
In an orthogonal coordinate system are equal to
Using the double argument identities for hyperbolic functions and trigonometric functions, the scale factors can be equivalently expressed as
Consequently, an infinitesimal element of area equals
and the Laplacian reads
by substituting the scale factors into the general formulae found in orthogonal coordinates.
Alternative definition
coordinate must be greater than or equal to one.
. For any point in the plane, the sum, whereas their difference, respectively.)
A drawback of these coordinates is that the points with Cartesian coordinates, so the conversion to Cartesian coordinates is not a function, but a multifunction.
Alternative scale factors
are
Hence, the infinitesimal area element becomes
and the Laplacian equals
by substituting the scale factors into the general formulae found in orthogonal coordinates.
Extrapolation to higher dimensions
Elliptic coordinates form the basis for several sets of three-dimensional orthogonal coordinates:
The elliptic cylindrical coordinates-direction.
The prolate spheroidal coordinates-axis, i.e., the axis connecting the foci, whereas the oblate spheroidal coordinates-axis, i.e., the axis separating the foci.
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.
Applications
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.
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).
See also
References
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