Elliptic cylindrical coordinates - AbsoluteAstronomy.com

Elliptic cylindrical coordinates are a three-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...

that results from projecting the two-dimensional elliptic coordinate system

Elliptic coordinates

In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in whichthe coordinate lines are confocal ellipses and hyperbolae...

in the
perpendicular

-direction. Hence, the coordinate surfaces are prisms

Prism (geometry)

In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

of 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 cylindrical coordinates

where

is a nonnegative real number and

.

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 cylindrical coordinates

and

are equal

whereas the remaining scale factor

.
Consequently, an infinitesimal volume element 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 (x,y) 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

and, of course,

. Hence, the infinitesimal volume 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

Applications

The classic applications of elliptic cylindrical 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 cylindrical coordinates allow 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....

. A typical example would be the electric field

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

surrounding a
flat conducting plate of width

.

The three-dimensional wave equation

Wave equation

The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...

, when expressed in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equations.

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).

External links

MathWorld description of elliptic cylindrical coordinates

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