Toroidal coordinates

The most common definition of toroidal coordinates $(\backslash tau,\; \backslash sigma,\; \backslash phi)$ is

:$$

x = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma} \cos \phi

:$$

y = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma} \sin \phi

:$$

z = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma}

together with $\backslash mathrm\{sign\}(\backslash sigma)=\backslash mathrm\{sign\}(z$).

The $\backslash sigma$ coordinate of a point $P$ equals the angle $F\_\{1\}\; P\; F\_\{2\}$ and the $\backslash tau$ coordinate equals the natural logarithm of the ratio of the distances $d\_\{1\}$ and $d\_\{2\}$ to opposite sides of the focal ring

:$$

\tau = \ln \frac{d_{1}}{d_{2}}.

The coordinate ranges are , $\backslash tau\backslash ge\; 0$ and

Image:Apollonian circles.svg about the vertical axis produces the three-dimensional toroidal coordinate system above. A circle on the vertical axis becomes the red sphere, whereas a circle on the horizontal axis becomes the blue torus.]]

Surfaces of constant $\backslash sigma$ correspond to spheres of different radii

:$$

\left( x^{2} + y^{2} \right) +

\left( z - a \cot \sigma \right)^{2} = \frac{a^{2}}{\sin^{2} \sigma}

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

:$$

z^{2} +

\left( \sqrt{x^{2} + y^{2}} - a \coth \tau \right)^{2} = \frac{a^{2}}{\sinh^{2} \tau}

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

The $(\backslash sigma,\; \backslash tau,\; \backslash phi)$ coordinates may be calculated from the Cartesian coordinates (x, y, z) as follows. The azimuthal angle $\backslash phi$ is given by the formula

:$$

\tan \phi = \frac{y}{x}

The cylindrical radius $\backslash rho$ of the point P is given by

:$$

\rho^{2} = x^{2} + y^{2} = \left(a \frac{\sinh \tau}{\cosh \tau - \cos \sigma}\right)^{2}

and its distances to the foci in the plane defined by $\backslash phi$ is given by

:$$

d_{1}^{2} = (\rho + a)^{2} + z^{2}

:$$

d_{2}^{2} = (\rho - a)^{2} + z^{2}

Image:Bipolar_coordinates.svg. The angle $\backslash sigma$ is formed by the two foci in this plane and P, whereas $\backslash tau$ is the logarithm of the ratio of distances to the foci. The corresponding circles of constant $\backslash sigma$ and $\backslash tau$ are shown in red and blue, respectively, and meet at right angles (magenta box); they are orthogonal.]]

The coordinate $\backslash tau$ equals the natural logarithm of the focal distances

:$$

\tau = \ln \frac{d_{1}}{d_{2}}

whereas $|\backslash sigma|$ equals the angle between the rays to the foci, which may be determined from the law of cosines

:$$

\cos \sigma = \frac{ d_{1}^{2} + d_{2}^{2} - 4 a^{2} }{2 d_{1} d_{2}}.

Or explicitly, including the sign,

:$$

\sigma = \mathrm{sign}(z)\arccos \frac{r^2-a^2}{\sqrt{(r^2-a^2)^2+4a^2z^2}}

where $r=\backslash sqrt\{\backslash rho^2+z^2\}$.

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

:$$

z+i\rho \ = ia\coth\frac{\tau+i\sigma}{2} ,

:$$

\tau+i\sigma \ = \ln\frac{ z+i(\rho+a) }{z+i(\rho-a)}.

The scale factors for the toroidal coordinates $\backslash sigma$ and $\backslash tau$ are equal

:$$

h_\sigma = h_\tau = \frac{a}{\cosh \tau - \cos\sigma}

whereas the azimuthal scale factor equals

:$$

h_\phi = \frac{a \sinh \tau}{\cosh \tau - \cos\sigma}

Thus, the infinitesimal volume element equals

:$$

dV = \frac{a^3 \sinh \tau}{\left( \cosh \tau - \cos\sigma \right)^3} \, d\sigma \, d\tau \, d\phi

The Laplacian is given by

\begin{align}

\nabla^2 \Phi =

\frac{\left( \cosh \tau - \cos\sigma \right)^{3}}{a^{2}\sinh \tau}

& \left[

\sinh \tau

\frac{\partial}{\partial \sigma}

\left( \frac{1}{\cosh \tau - \cos\sigma}

\frac{\partial \Phi}{\partial \sigma}

\right) \right. \\[8pt]

& {} \quad +

\left. \frac{\partial}{\partial \tau}

\left( \frac{\sinh \tau}{\cosh \tau - \cos\sigma}

\frac{\partial \Phi}{\partial \tau}

\right) +

\frac{1}{\sinh \tau \left( \cosh \tau - \cos\sigma \right)}

\frac{\partial^2 \Phi}{\partial \phi^2}

\right]

\end{align}

For a vector field $$\backslash vec\{n\}(\backslash tau,\backslash sigma,\backslash phi)\; =\; n\_\{\backslash tau\}(\backslash tau,\backslash sigma,\backslash phi)\backslash hat\{e\}\_\{\backslash tau\}\; +\; n\_\{\backslash sigma\}(\backslash tau,\backslash sigma,\backslash phi)\; \backslash hat\{e\}\_\{\backslash sigma\}\; +\; n\_\{\backslash phi\}\; (\backslash tau,\backslash sigma,\backslash phi)\; \backslash hat\{e\}\_\{\backslash phi\},$$ the Vector Laplacian is given by

$$\backslash begin\{align\}$$

\Delta \vec{n}(\tau,\sigma,\phi) &= \nabla (\nabla \cdot \vec{n}) - \nabla \times (\nabla \times \vec{n}) \\

&= \frac{1}{a^2}\vec{e}_{\tau} \left \{

n_{\tau} \left( -\frac{\sinh^4 \tau + (\cosh \tau - \cos \sigma)^2}{\sinh^2 \tau} \right)

+ n_{\sigma} (- \sinh \tau \sin \sigma )

+ \frac{\partial n_{\tau}}{\partial \tau} \left( \frac{(\cosh \tau - \cos \sigma)(1 - \cosh \tau \cos \sigma)}{\sinh \tau} \right) + \cdots \right. \\

&\qquad + \frac{\partial n_{\tau}}{\partial \sigma} ( -(\cosh \tau - \cos \sigma) \sin \sigma )

+ \frac{\partial n_{\sigma}}{\partial \sigma} ( 2(\cosh \tau - \cos \sigma) \sinh \tau )

+ \frac{\partial n_{\sigma}}{\partial \tau} ( -2(\cosh \tau - \cos \sigma) \sin \sigma ) + \cdots \\

&\qquad + \frac{\partial n_{\phi}}{\partial \phi} \left( \frac{-2(\cosh \tau - \cos \sigma)(1 - \cosh \tau \cos \sigma)}{\sinh^2 \tau} \right)

+ \frac{\partial^2 n_{\tau}}{{\partial \tau}^2} (\cosh \tau - \cos \sigma)^2

+ \frac{\partial^2 n_{\tau}}{{\partial \sigma}^2} (- (\cosh \tau - \cos \sigma)^2 ) + \cdots \\

& \qquad \left. + \frac{\partial^2 n_{\tau}}{{\partial \phi}^2} \frac{(\cosh \tau - \cos \sigma)^2}{\sinh^2 \tau}

\right \}\\

&+ \frac{1}{a^2}\vec{e}_{\sigma} \left \{

n_{\tau} \left( -\frac{(\cosh^2 \tau + 1 -2\cosh \tau \cos \sigma)\sin \sigma}{\sinh \tau} \right)

+ n_{\sigma} \left( - \sinh^2 \tau - 2\sin^2 \sigma \right) + \ldots \right.\\

&\qquad \left. + \frac{\partial n_{\tau}}{\partial \tau} (2 \sin \sigma (\cosh \tau - \cos \sigma) )

+ \frac{\partial n_{\tau}}{\partial \sigma} \left( -2 \sinh \tau (\cosh \tau - \cos \sigma) \right) + \cdots \right.\\

&\qquad \left. + \frac{\partial n_{\sigma}}{\partial \tau} \left( \frac{(\cosh \tau - \cos \sigma) (1 - \cosh \tau \cos \sigma) }{\sinh \tau} \right)

+ \frac{\partial n_{\sigma}}{\partial \sigma} ( -(\cosh \tau - \cos \sigma)\sin \sigma) + \cdots \right.\\

&\qquad \left. + \frac{\partial n_{\phi}}{\partial \phi} \left( 2\frac{(\cosh \tau - \cos \sigma)\sin \sigma}{\sinh \tau} \right)

+ \frac{\partial^2 n_{\sigma}}{{\partial \tau}^2} (\cosh \tau - \cos \sigma)^2

+ \frac{\partial^2 n_{\sigma}}{{\partial \sigma}^2} (\cosh \tau - \cos \sigma)^2 + \cdots \right.\\

&\qquad \left. + \frac{\partial^2 n_{\sigma}}{{\partial \phi}^2} \left( \frac{(\cosh \tau - \cos \sigma)^2}{\sinh^2 \tau} \right)

\right \}\\

&+ \frac{1}{a^2}\vec{e}_{\phi} \left \{

n_{\phi} \left( -\frac{(\cosh \tau - \cos \sigma)^2}{\sinh^2 \tau} \right)

+ \frac{\partial n_{\tau}}{\partial \phi} \left( \frac{2(\cosh \tau - \cos \sigma)(1 - \cosh \tau \cos \sigma)}{\sinh^2 \tau} \right) + \cdots \right.\\

&\qquad \left. + \frac{\partial n_{\sigma}}{\partial \phi} \left( -\frac{2(\cosh \tau - \cos \sigma) \sin \sigma}{\sinh \tau} \right)

+ \frac{\partial n_{\phi}}{\partial \tau} \left( \frac{(\cosh \tau - \cos \sigma)(1 - \cosh \tau \cos \sigma)}{\sinh \tau} \right) + \cdots \right.\\

&\qquad \left. + \frac{\partial n_{\phi}}{\partial \sigma} (-(\cosh \tau - \cos \sigma) \sin \sigma ) + \frac{\partial^2 n_{\phi}}{{\partial \tau}^2} (\cosh \tau - \cos \sigma)^2 + \cdots \right. \\

&\qquad \left. + \frac{\partial^2 n_{\phi}}{{\partial \sigma}^2} (\cosh \tau - \cos \sigma)^2

+ \frac{\partial^2 n_{\phi}}{{\partial \phi}^2} \left( \frac{(\cosh \tau - \cos \sigma)^2}{\sinh^2 \tau} \right)

\right \}

\end{align}

Other differential operators such as $\backslash nabla\; \backslash cdot\; \backslash mathbf\{F\}$

and $\backslash nabla\; \backslash times\; \backslash mathbf\{F\}$ can be expressed in the coordinates $(\backslash sigma,\; \backslash tau,\; \backslash phi)$ by substituting

the scale factors into the general formulae

found in orthogonal coordinates.

The 3-variable Laplace equation

:$\backslash nabla^2\backslash Phi=0$

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

:$$

\Phi=U\sqrt{\cosh\tau-\cos\sigma}

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

:$\backslash Phi=\; \backslash sqrt\{\backslash cosh\backslash tau-\backslash cos\backslash sigma\}\backslash ,\backslash ,S\_\backslash nu(\backslash sigma)T\_\{\backslash mu\backslash nu\}(\backslash tau)V\_\backslash mu(\backslash phi)$

where each function is a linear combination of:

:$$

S_\nu(\sigma)=e^{i\nu\sigma}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\nu\sigma}

:$$

T_{\mu\nu}(\tau)=P_{\nu-1/2}^\mu(\cosh\tau)\,\,\,\,\mathrm{and}\,\,\,\,Q_{\nu-1/2}^\mu(\cosh\tau)

:$$

V_\mu(\phi)=e^{i\mu\phi}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\mu\phi}

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 $z=\backslash cosh\backslash tau>1$ then, for instance, with vanishing order $\backslash mu=0$ (the convention is to not write the order when it vanishes) and $\backslash nu=0$

:$Q\_\{-\backslash frac12\}(z)=\backslash sqrt\{\backslash frac\{2\}\{1+z\}\}K\backslash left(\backslash sqrt\{\backslash frac\{2\}\{1+z\}\}\backslash right)$

and

:$P\_\{-\backslash frac12\}(z)=\backslash frac\{2\}\{\backslash pi\}\backslash sqrt\{\backslash frac\{2\}\{1+z\}\}K\; \backslash left(\; \backslash sqrt\{\backslash frac\{z-1\}\{z+1\}\}\; \backslash right)$

where $\backslash ,\backslash !K$ and $\backslash ,\backslash !E$ 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 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).

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

:$$

\Phi=\frac{U}{\sqrt{\rho}}

where

:$$

\rho=\sqrt{x^2+y^2}=\frac{a\sinh\tau}{\cosh\tau-\cos\sigma}.

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

:$\backslash Phi=\; \backslash frac\{a\}\{\backslash sqrt\{\backslash rho\}\}\backslash ,\backslash ,S\_\backslash nu(\backslash sigma)T\_\{\backslash mu\backslash nu\}(\backslash tau)V\_\backslash mu(\backslash phi)$

where each function is a linear combination of:

:$$

S_\nu(\sigma)=e^{i\nu\sigma}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\nu\sigma}

:$$

T_{\mu\nu}(\tau)=P_{\mu-1/2}^\nu(\coth\tau)\,\,\,\,\mathrm{and}\,\,\,\,Q_{\mu-1/2}^\nu(\coth\tau)

:$$

V_\mu(\phi)=e^{i\mu\phi}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\mu\phi}.

Note that although the toroidal harmonics are used again for the T  function, the argument is $\backslash coth\backslash tau$ rather than $\backslash cosh\backslash tau$ and the $\backslash mu$ and $\backslash nu$ indices are exchanged. This method is useful for situations in which the boundary conditions are independent of the spherical angle $\backslash theta$, 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.

• Byerly, W E. (1893) [https://archive.org/details/elemtreatfour00byerrich An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics] Ginn & co. pp. 264–266
• {{cite book | author = Arfken G | year = 1970 | title = Mathematical Methods for Physicists | edition = 2nd | publisher = Academic Press | location = Orlando, FL | pages = 112–115}}
• {{cite journal |last=Andrews |first=Mark |year=2006 |title=Alternative separation of Laplace's equation in toroidal coordinates and its application to electrostatics |journal=Journal of Electrostatics |volume=64|pages=664–672 |doi=10.1016/j.elstat.2005.11.005 |issue=10|citeseerx=10.1.1.205.5658 }}
• {{cite journal |last=Hulme |first=A. |year=1982 |title=A note on the magnetic scalar potential of an electric current-ring |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=92|pages=183–191 |doi=10.1017/S0305004100059831|issue=1|bibcode=1982MPCPS..92..183H }}

• {{cite book | author = Morse P M, Feshbach H | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw–Hill | location = New York | page = 666}}
• {{cite book | author = Korn G A, Korn T M |year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | page = 182 | lccn = 59014456}}
• {{cite book | author = Margenau H, Murphy G M | year = 1956 | title = The Mathematics of Physics and Chemistry | url = https://archive.org/details/mathematicsphysi00marg_501 | url-access = limited | publisher = D. van Nostrand | location = New York| pages = [https://archive.org/details/mathematicsphysi00marg_501/page/n203 190]–192 | lccn = 55010911 }}
• {{cite book | author = Moon P H, Spencer D E | year = 1988 | chapter = Toroidal Coordinates (η, θ, ψ) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = 2nd ed., 3rd revised printing | publisher = Springer Verlag | location = New York | isbn = 978-0-387-02732-6 | pages = 112–115 (Section IV, E4Ry)}}

• [http://mathworld.wolfram.com/ToroidalCoordinates.html MathWorld description of toroidal coordinates]

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