Elliptic coordinates - AbsoluteAstronomy.com

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the elliptic coordinate system is a two-dimensional orthogonal

Orthogonal coordinates

In mathematics, orthogonal coordinates are defined as a set of d coordinates q = in which the coordinate surfaces all meet at right angles . A coordinate surface for a particular coordinate qk is the curve, surface, or hypersurface on which qk is a constant...

coordinate system

Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

in which
the coordinate lines are confocal ellipse

Ellipse

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

s and hyperbola

Hyperbola

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

e. The two foci

Focus (geometry)

In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...

and

are generally taken to be fixed at

and

, respectively, on the

-axis of the Cartesian coordinate system

Cartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

Basic definition

The most common definition of elliptic coordinates

where

is a nonnegative real number and

On the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

, an equivalent relationship is

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

shows that curves of constant

form ellipse

Ellipse

s, whereas the hyperbolic trigonometric identity

shows that curves of constant

form hyperbola

Hyperbola

Scale factors

The scale factors for the elliptic coordinates

are equal

To simplify the computation of the scale factors, double angle identities can be used to express them equivalently as

Consequently, an infinitesimal element of area equals

and the Laplacian equals

Other differential operators such as

and

can be expressed in the coordinates

by substituting
the scale factors into the general formulae found in orthogonal coordinates

Orthogonal coordinates

Alternative definition

An alternative and geometrically intuitive set of elliptic coordinates

are sometimes used,
where

and

. Hence, the curves of constant

are ellipses, whereas the curves of constant

are hyperbolae. The coordinate

must belong to the interval [-1, 1], whereas the

coordinate must be greater than or equal to one.

The coordinates

have a simple relation to the distances to the foci

and

. For any point in the plane, the sum

of its distances to the foci equals

, whereas their difference

equals

.
Thus, the distance to

, whereas the distance to

. (Recall that

and

are located at

and

, respectively.)

A drawback of these coordinates is that they do not have a 1-to-1 transformation to the Cartesian coordinates

Alternative scale factors

The scale factors for the alternative elliptic coordinates

are

Hence, the infinitesimal area element becomes

and the Laplacian equals

Other differential operators such as

and

can be expressed in the coordinates

by substituting
the scale factors into the general formulae
found in orthogonal coordinates

Orthogonal coordinates

Extrapolation to higher dimensions

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

Orthogonal coordinates

.
The elliptic cylindrical coordinates

Elliptic cylindrical coordinates

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the...

are produced by projecting in the

-direction.
The prolate spheroidal coordinates

Prolate spheroidal coordinates

Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating a spheroid around its major axis, i.e., the axis on which the foci are located...

are produced by rotating the elliptic coordinates about the

-axis, i.e., the axis connecting the foci, whereas the oblate spheroidal coordinates

Oblate spheroidal coordinates

Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of...

are produced by rotating the elliptic coordinates about the

-axis, i.e., the axis separating the foci.

Applications

The classic applications of elliptic coordinates are in solving partial differential equations,
e.g., Laplace's equation

Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

or the Helmholtz equation

Helmholtz equation

The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation\nabla^2 A + k^2 A = 0where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...

, for which elliptic coordinates are a natural description of a system thus allowing a separation of variables

Separation of variables

In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....

in the partial differential equations. Some traditional examples are solving systems such as electrons orbiting a molecule or planetary orbits that have a elliptical shape.

The geometric properties of elliptic coordinates can also be useful. A typical example might involve
an integration over all pairs of vectors

and

that sum to a fixed vector

, where the integrand
was a function of the vector lengths

and

. (In such a case, one would position

between the two foci and aligned with the

-axis, i.e.,

.) For concreteness,

and

could represent the momenta

Momentum

In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

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).

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