Toroidal coordinates

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 F_{1} and F_{2} in bipolar coordinates become a ring of radius a in the xy plane of the toroidal coordinate system; the z-axis is the axis of rotation. The focal ring is also known as the reference circle.


The most common definition of toroidal coordinates (sigma, au, phi) is

:x = a frac{sinh au}{cosh au - cos sigma} cos phi

:y = a frac{sinh au}{cosh au - cos sigma} sin phi

:z = a frac{sin sigma}{cosh au - cos sigma}

where the sigma coordinate of a point P equals the angle F_{1} P F_{2} and the au coordinate equals the natural logarithm of the ratio of the distances d_{1} and d_{2} to opposite sides of the focal ring

: au = ln frac{d_{1{d_{2.

The coordinate ranges are -pi and auge 0 and 0lephi < 2pi.

Coordinate surfaces

Surfaces of constant sigma correspond to spheres of different radii

:left( x^{2} + y^{2} ight) +left( z - a cot sigma ight)^{2} = frac{a^{2{sin^{2} sigma}

that all pass through the focal ring but are not concentric. The surfaces of constant au are non-intersecting tori of different radii

:z^{2} +left( sqrt{x^{2} + y^{2 - a coth au ight)^{2} = frac{a^{2{sinh^{2} au}

that surround the focal ring. The centers of the constant-sigma spheres lie along the z-axis, whereas the constant- au tori are centered in the xy 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

: an phi = frac{y}{x}

The cylindrical radius ρ of the point P is given by

: ho^{2} = x^{2} + y^{2}

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

:d_{1}^{2} = ( ho + a)^{2} + z^{2}

:d_{2}^{2} = ( ho - a)^{2} + z^{2}

The coordinate τ equals the natural logarithm of the focal distances

: au = ln frac{d_{1{d_{2

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

:cos sigma = -frac{4a^{2} - d_{1}^{2} - d_{2}^{2{2 d_{1} d_{2

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

cale factors

The scale factors for the toroidal coordinates sigma and au are equal

:h_{sigma} = h_{ au} = frac{a}{cosh au - cossigma}

whereas the azimuthal scale factor equals

:h_{phi} = frac{a sinh au}{cosh au - cossigma}

Thus, the infinitesimal volume element equals

:dV= frac{a^{3}sinh au}{left( cosh au - cossigma ight)^{3 dsigma d au dphi

and the Laplacian is given by

: abla^{2} Phi =frac{left( cosh au - cossigma ight)^{3{a^{2}sinh au} left [ sinh au frac{partial}{partial sigma}left( frac{1}{cosh au - cossigma}frac{partial Phi}{partial sigma} ight) + frac{partial}{partial au}left( frac{sinh au}{cosh au - cossigma}frac{partial Phi}{partial au} ight) + frac{1}{sinh au left( cosh au - cossigma ight)}frac{partial^{2} Phi}{partial phi^{2 ight]

Other differential operators such as abla cdot mathbf{F} and abla imes mathbf{F} can be expressed in the coordinates (sigma, au, phi) by substituting the scale factors into the general formulae found in orthogonal coordinates.

Toroidal Harmonics

tandard separation

The 3-variable Laplace equation

: abla^2Psi=0

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

:V=Usqrt{cosh au-cossigma}

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

:V= sqrt{cosh au-cossigma},,S_ u(sigma)T_{mu u}( au)Phi_mu(phi),

where each function is a linear combination of:

:S_ u(sigma)=e^{i usigma},,,,mathrm{and},,,,e^{-i usigma}:T_{mu u}( au)=P_{ u-1/2}^mu(cosh au),,,,mathrm{and},,,,Q_{ u-1/2}^mu(cosh au):Phi_mu(phi)=e^{imuphi},,,,mathrm{and},,,,e^{-imuphi}

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 ,!1 then, for instance, with vanishing order (the convention is to not write the order when it vanishes) and ,!n=0

:Q_{-frac12}(z)=sqrt{frac{2}{1+zKleft(sqrt{frac{2}{1+z ight)


:P_{-frac12}(z)=frac{2}{pi}sqrt{frac{2}{1+zK left( sqrt{frac{z-1}{z+1 ight)

where ,!K and ,!E 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 does 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, a conducting ring.

An alternative separation

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

:V=frac{U}{sqrt{ ho


: ho=sqrt{x^2+y^2}=frac{cosh au-cossigma}{asinh au}

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

:V= frac{a}{ ho},,S_ u(sigma)T_{mu u}( au)Phi_mu(phi),

where each function is a linear combination of:

:S_ u(sigma)=e^{i usigma},,,,mathrm{and},,,,e^{-i usigma}:T_{mu u}( au)=P_{mu-1/2}^ u(coth au),,,,mathrm{and},,,,Q_{mu-1/2}^ u(coth au):Phi_mu(phi)=e^{imuphi},,,,mathrm{and},,,,e^{-imuphi}

Note that although the toroidal harmonics are used again for the "T" function, the argument is coth au rather than cosh au and the mu and u indices are exchanged. This method is useful for situations in which the boundary conditions are independent of the spherical angle heta, such as the charged ring, an infinite half plane, or two parallel planes. For identities relating the toroidal harmonics with argument hyperboliccosine with those of argument hyperbolic cotangent, see the Whipple formulae.

See also


* Byerly, WE. (1893) " [ An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics] " Ginn &amp; co. pp. 264-266





* | pages = p. 182

*| pages = pp. 190&ndash;192


External links

* [ MathWorld description of toroidal coordinates]

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