Toroidal coordinates - AbsoluteAstronomy.com

Toroidal 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 rotating the two-dimensional bipolar coordinate system

Bipolar coordinates

Bipolar coordinates are a two-dimensional orthogonal coordinate system. There are two commonly defined types of bipolar coordinates. The other system is two-center bipolar coordinates. There is also a third coordinate system that is based on two poles . The first is based on the Apollonian circles...

about the axis that separates its two foci. Thus, 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

in bipolar coordinates

Bipolar coordinates

become a ring of radius

in the

plane of the toroidal coordinate system; the

-axis is the axis of rotation. The focal ring is also known as the reference circle.

Definition

The most common definition of toroidal coordinates

where the

coordinate of a point

equals the angle

and the

coordinate equals the natural logarithm

Natural logarithm

The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

of the ratio of the distances

and

to opposite sides of the focal ring

The coordinate ranges are

and

Coordinate surfaces

Surfaces of constant

correspond to spheres of different radii

that all pass through the focal ring but are not concentric. The surfaces of constant

are non-intersecting tori of different radii

that surround the focal ring. The centers of the constant-

spheres lie along the

-axis, whereas the constant-

tori are centered in the

plane.

Inverse transformation

The (σ, τ, φ) coordinates may be calculated from the Cartesian coordinates (x, y, z) as follows. The azimuthal angle φ is given by the formula

The cylindrical radius ρ of the point P is given by

and its distances to the foci in the plane defined by φ is given by

The coordinate τ equals the natural logarithm

Natural logarithm

The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

of the focal distances

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

Law of cosines

In trigonometry, the law of cosines relates the lengths of the sides of a plane triangle to the cosine of one of its angles. Using notation as in Fig...

where the sign of σ is determined by whether the coordinate surface sphere is above or below the x-y plane.

Scale factors

The scale factors for the toroidal coordinates

and

are equal

whereas the azimuthal scale factor equals

Thus, the infinitesimal volume element equals

and the Laplacian is given by

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

Standard separation

The 3-variable Laplace equation

admits solution via 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 toroidal coordinates. Making the substitution

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

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.

Toroidal harmonics have many interesting properties. If you make a variable substitution

then, for instance, with vanishing order (the convention is to not write the order when it vanishes) and

and

where

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

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...

for which toroidal coordinates allow a separation of variables

Separation of variables

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 toroidal coordinates do not allow a separation of variables. Typical examples would be the electric potential

Electric potential

In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

and 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...

of a conducting torus, or in the degenerate case, a conducting ring.

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

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 function, the argument is

rather than

and the

and

indices are exchanged. This method is useful for situations in which the boundary conditions are independent of the spherical angle

, 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

Whipple formulae

In the theory of special functions, Whipple's transformation for Legendre functions, named after Francis John Welsh Whipple, arise from a general expression, concerning associated Legendre functions...

External links

MathWorld description of toroidal coordinates

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