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Wikipedia / Toroidal coordinates

Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci in bipolar coordinates-axis is the axis of rotation. The focal ring is also known as the reference circle.

Definition

is

coordinate equals the natural logarithm to opposite sides of the focal ring

Coordinate surfaces

correspond to spheres of different radii

are non-intersecting tori of different radii

plane.

Inverse transformation

coordinates may be calculated from the Cartesian coordinates (x, y, z is given by the formula

of the point P is given by

is given by

equals the natural logarithm of the focal distances

equals the angle between the rays to the foci, which may be determined from the law of cosines

Or explicitly, including the sign,

.

The transformations between cylindrical and toroidal coordinates can be expressed in complex notation as

Scale factors

are equal

whereas the azimuthal scale factor equals

Thus, the infinitesimal volume element equals

Differential Operators

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

Toroidal harmonics

Standard separation

The 3-variable Laplace equation

admits solution via separation of variables in toroidal coordinates. Making the substitution

A separable equation is then obtained. A particular solution obtained by separation of variables is:

where each function is a linear combination of:

Where P and Q are associated Legendre functions of the first and second kind. These Legendre functions are often referred to as toroidal harmonics.

and

are the complete elliptic integrals of the first and second kind respectively. The rest of the toroidal harmonics can be obtained, for instance, in terms of the complete elliptic integrals, by using recurrence relations for associated Legendre functions.

The classic applications of toroidal coordinates are in solving partial differential equations, e.g., Laplace's equation for which toroidal coordinates allow a separation of variables or the Helmholtz equation, for which toroidal coordinates do not allow a separation of variables. Typical examples would be the electric potential and electric field of a conducting torus, or in the degenerate case, an electric current-ring (Hulme 1982).

An alternative separation

Alternatively, a different substitution may be made (Andrews 2006)

where

Again, a separable equation is obtained. A particular solution obtained by separation of variables is then:

where each function is a linear combination of:

Note that although the toroidal harmonics are used again for the T, such as the charged ring, an infinite half plane, or two parallel planes. For identities relating the toroidal harmonics with argument hyperbolic cosine with those of argument hyperbolic cotangent, see the Whipple formulae.

References

Bibliography