# Toroidal coordinates

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Three-dimensional orthogonal coordinate system

Illustration of toroidal coordinates, which are obtained by rotating a two-dimensional [bipolar coordinate system](/source/Bipolar_coordinates) about the axis separating its two foci.  The foci are located at a distance 1 from the vertical *z*-axis.  The portion of the red sphere that lies above the

        x
        y

    {\displaystyle xy}

-plane is the σ = 30° isosurface, the blue torus is the τ = 0.5 isosurface, and the yellow half-plane is the φ = 60° isosurface.  The green half-plane marks the *x*-*z* plane, from which φ is measured.  The black point is located at the intersection of the red, blue and yellow isosurfaces, at Cartesian coordinates roughly (0.996, −1.725, 1.911).

**Toroidal coordinates** are a three-dimensional [orthogonal](/source/Orthogonal_coordinates) [coordinate system](/source/Coordinate_system) that results from rotating the two-dimensional [bipolar coordinate system](/source/Bipolar_coordinates) about the axis that separates its two foci. Thus, the two [foci](/source/Focus_(geometry)) F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} in [bipolar coordinates](/source/Bipolar_coordinates) become a ring of radius a {\displaystyle a} in the x y {\displaystyle xy} plane of the toroidal coordinate system; the z {\displaystyle z} -axis is the axis of rotation. The focal ring is also known as the reference circle.

## Definition

The most common definition of toroidal coordinates ( τ , σ , ϕ ) {\displaystyle (\tau ,\sigma ,\phi )} is

- x = a sinh ⁡ τ cosh ⁡ τ − cos ⁡ σ cos ⁡ ϕ {\displaystyle x=a\ {\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}\cos \phi }

- y = a sinh ⁡ τ cosh ⁡ τ − cos ⁡ σ sin ⁡ ϕ {\displaystyle y=a\ {\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}\sin \phi }

- z = a sin ⁡ σ cosh ⁡ τ − cos ⁡ σ {\displaystyle z=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}}

together with s i g n ( σ ) = s i g n ( z {\displaystyle \mathrm {sign} (\sigma )=\mathrm {sign} (z} ). The σ {\displaystyle \sigma } coordinate of a point P {\displaystyle P} equals the angle F 1 P F 2 {\displaystyle F_{1}PF_{2}} and the τ {\displaystyle \tau } coordinate equals the [natural logarithm](/source/Natural_logarithm) of the ratio of the distances d 1 {\displaystyle d_{1}} and d 2 {\displaystyle d_{2}} to opposite sides of the focal ring

- τ = ln ⁡ d 1 d 2 . {\displaystyle \tau =\ln {\frac {d_{1}}{d_{2}}}.}

The coordinate ranges are − π < σ ≤ π {\displaystyle -\pi <\sigma \leq \pi } , τ ≥ 0 {\displaystyle \tau \geq 0} and 0 ≤ ϕ < 2 π . {\displaystyle 0\leq \phi <2\pi .}

### Coordinate surfaces

Rotating this two-dimensional [bipolar coordinate system](/source/Bipolar_coordinates) about the vertical axis produces the three-dimensional toroidal coordinate system above.  A circle on the vertical axis becomes the red [sphere](/source/Sphere), whereas a circle on the horizontal axis becomes the blue [torus](/source/Torus).

Surfaces of constant σ {\displaystyle \sigma } correspond to spheres of different radii

- ( x 2 + y 2 ) + ( z − a cot ⁡ σ ) 2 = a 2 sin 2 ⁡ σ {\displaystyle \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 τ {\displaystyle \tau } are non-intersecting tori of different radii

- z 2 + ( x 2 + y 2 − a coth ⁡ τ ) 2 = a 2 sinh 2 ⁡ τ {\displaystyle 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- σ {\displaystyle \sigma } spheres lie along the z {\displaystyle z} -axis, whereas the constant- τ {\displaystyle \tau } tori are centered in the x y {\displaystyle xy} plane.

### Inverse transformation

The ( σ , τ , ϕ ) {\displaystyle (\sigma ,\tau ,\phi )} coordinates may be calculated from the Cartesian coordinates (*x*, *y*, *z*) as follows. The azimuthal angle ϕ {\displaystyle \phi } is given by the formula

- tan ⁡ ϕ = y x {\displaystyle \tan \phi ={\frac {y}{x}}}

The cylindrical radius ρ {\displaystyle \rho } of the point P is given by

- ρ 2 = x 2 + y 2 = ( a sinh ⁡ τ cosh ⁡ τ − cos ⁡ σ ) 2 {\displaystyle \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 ϕ {\displaystyle \phi } is given by

- d 1 2 = ( ρ + a ) 2 + z 2 {\displaystyle d_{1}^{2}=(\rho +a)^{2}+z^{2}}

- d 2 2 = ( ρ − a ) 2 + z 2 {\displaystyle d_{2}^{2}=(\rho -a)^{2}+z^{2}}

Geometric interpretation of the coordinates σ and τ of a point **P**.  Observed in the plane of constant azimuthal angle

        ϕ

    {\displaystyle \phi }

, toroidal coordinates are equivalent to [bipolar coordinates](/source/Bipolar_coordinates).  The angle

        σ

    {\displaystyle \sigma }

 is formed by the two foci in this plane and **P**, whereas

        τ

    {\displaystyle \tau }

 is the logarithm of the ratio of distances to the foci.  The corresponding circles of constant

        σ

    {\displaystyle \sigma }

 and

        τ

    {\displaystyle \tau }

 are shown in red and blue, respectively, and meet at right angles (magenta box); they are orthogonal.

The coordinate τ {\displaystyle \tau } equals the [natural logarithm](/source/Natural_logarithm) of the focal distances

- τ = ln ⁡ d 1 d 2 {\displaystyle \tau =\ln {\frac {d_{1}}{d_{2}}}}

whereas | σ | {\displaystyle |\sigma |} equals the angle between the rays to the foci, which may be determined from the [law of cosines](/source/Law_of_cosines)

- cos ⁡ σ = d 1 2 + d 2 2 − 4 a 2 2 d 1 d 2 . {\displaystyle \cos \sigma ={\frac {d_{1}^{2}+d_{2}^{2}-4a^{2}}{2d_{1}d_{2}}}.}

Or explicitly, including the sign,

- σ = s i g n ( z ) arccos ⁡ r 2 − a 2 ( r 2 − a 2 ) 2 + 4 a 2 z 2 {\displaystyle \sigma =\mathrm {sign} (z)\arccos {\frac {r^{2}-a^{2}}{\sqrt {(r^{2}-a^{2})^{2}+4a^{2}z^{2}}}}}

where r = ρ 2 + z 2 {\displaystyle r={\sqrt {\rho ^{2}+z^{2}}}} .

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

- z + i ρ = i a coth ⁡ τ + i σ 2 , {\displaystyle z+i\rho \ =ia\coth {\frac {\tau +i\sigma }{2}},}

- τ + i σ = ln ⁡ z + i ( ρ + a ) z + i ( ρ − a ) . {\displaystyle \tau +i\sigma \ =\ln {\frac {z+i(\rho +a)}{z+i(\rho -a)}}.}

### Scale factors

The scale factors for the toroidal coordinates σ {\displaystyle \sigma } and τ {\displaystyle \tau } are equal

- h σ = h τ = a cosh ⁡ τ − cos ⁡ σ {\displaystyle h_{\sigma }=h_{\tau }={\frac {a}{\cosh \tau -\cos \sigma }}}

whereas the azimuthal scale factor equals

- h ϕ = a sinh ⁡ τ cosh ⁡ τ − cos ⁡ σ {\displaystyle h_{\phi }={\frac {a\sinh \tau }{\cosh \tau -\cos \sigma }}}

Thus, the [infinitesimal](/source/Infinitesimal) [volume element](/source/Volume_element) equals

- d V = a 3 sinh ⁡ τ ( cosh ⁡ τ − cos ⁡ σ ) 3 d σ d τ d ϕ {\displaystyle dV={\frac {a^{3}\sinh \tau }{\left(\cosh \tau -\cos \sigma \right)^{3}}}\,d\sigma \,d\tau \,d\phi }

### Differential Operators

The Laplacian is given by ∇ 2 Φ = ( cosh ⁡ τ − cos ⁡ σ ) 3 a 2 sinh ⁡ τ [ sinh ⁡ τ ∂ ∂ σ ( 1 cosh ⁡ τ − cos ⁡ σ ∂ Φ ∂ σ ) + ∂ ∂ τ ( sinh ⁡ τ cosh ⁡ τ − cos ⁡ σ ∂ Φ ∂ τ ) + 1 sinh ⁡ τ ( cosh ⁡ τ − cos ⁡ σ ) ∂ 2 Φ ∂ ϕ 2 ] {\displaystyle {\begin{aligned}\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{aligned}}}

For a vector field n → ( τ , σ , ϕ ) = n τ ( τ , σ , ϕ ) e ^ τ + n σ ( τ , σ , ϕ ) e ^ σ + n ϕ ( τ , σ , ϕ ) e ^ ϕ , {\displaystyle {\vec {n}}(\tau ,\sigma ,\phi )=n_{\tau }(\tau ,\sigma ,\phi ){\hat {e}}_{\tau }+n_{\sigma }(\tau ,\sigma ,\phi ){\hat {e}}_{\sigma }+n_{\phi }(\tau ,\sigma ,\phi ){\hat {e}}_{\phi },} the Vector Laplacian is given by Δ n → ( τ , σ , ϕ ) = ∇ ( ∇ ⋅ n → ) − ∇ × ( ∇ × n → ) = 1 a 2 e → τ { n τ ( − sinh 4 ⁡ τ + ( cosh ⁡ τ − cos ⁡ σ ) 2 sinh 2 ⁡ τ ) + n σ ( − sinh ⁡ τ sin ⁡ σ ) + ∂ n τ ∂ τ ( ( cosh ⁡ τ − cos ⁡ σ ) ( 1 − cosh ⁡ τ cos ⁡ σ ) sinh ⁡ τ ) + ⋯ + ∂ n τ ∂ σ ( − ( cosh ⁡ τ − cos ⁡ σ ) sin ⁡ σ ) + ∂ n σ ∂ σ ( 2 ( cosh ⁡ τ − cos ⁡ σ ) sinh ⁡ τ ) + ∂ n σ ∂ τ ( − 2 ( cosh ⁡ τ − cos ⁡ σ ) sin ⁡ σ ) + ⋯ + ∂ n ϕ ∂ ϕ ( − 2 ( cosh ⁡ τ − cos ⁡ σ ) ( 1 − cosh ⁡ τ cos ⁡ σ ) sinh 2 ⁡ τ ) + ∂ 2 n τ ∂ τ 2 ( cosh ⁡ τ − cos ⁡ σ ) 2 + ∂ 2 n τ ∂ σ 2 ( − ( cosh ⁡ τ − cos ⁡ σ ) 2 ) + ⋯ + ∂ 2 n τ ∂ ϕ 2 ( cosh ⁡ τ − cos ⁡ σ ) 2 sinh 2 ⁡ τ } + 1 a 2 e → σ { n τ ( − ( cosh 2 ⁡ τ + 1 − 2 cosh ⁡ τ cos ⁡ σ ) sin ⁡ σ sinh ⁡ τ ) + n σ ( − sinh 2 ⁡ τ − 2 sin 2 ⁡ σ ) + … + ∂ n τ ∂ τ ( 2 sin ⁡ σ ( cosh ⁡ τ − cos ⁡ σ ) ) + ∂ n τ ∂ σ ( − 2 sinh ⁡ τ ( cosh ⁡ τ − cos ⁡ σ ) ) + ⋯ + ∂ n σ ∂ τ ( ( cosh ⁡ τ − cos ⁡ σ ) ( 1 − cosh ⁡ τ cos ⁡ σ ) sinh ⁡ τ ) + ∂ n σ ∂ σ ( − ( cosh ⁡ τ − cos ⁡ σ ) sin ⁡ σ ) + ⋯ + ∂ n ϕ ∂ ϕ ( 2 ( cosh ⁡ τ − cos ⁡ σ ) sin ⁡ σ sinh ⁡ τ ) + ∂ 2 n σ ∂ τ 2 ( cosh ⁡ τ − cos ⁡ σ ) 2 + ∂ 2 n σ ∂ σ 2 ( cosh ⁡ τ − cos ⁡ σ ) 2 + ⋯ + ∂ 2 n σ ∂ ϕ 2 ( ( cosh ⁡ τ − cos ⁡ σ ) 2 sinh 2 ⁡ τ ) } + 1 a 2 e → ϕ { n ϕ ( − ( cosh ⁡ τ − cos ⁡ σ ) 2 sinh 2 ⁡ τ ) + ∂ n τ ∂ ϕ ( 2 ( cosh ⁡ τ − cos ⁡ σ ) ( 1 − cosh ⁡ τ cos ⁡ σ ) sinh 2 ⁡ τ ) + ⋯ + ∂ n σ ∂ ϕ ( − 2 ( cosh ⁡ τ − cos ⁡ σ ) sin ⁡ σ sinh ⁡ τ ) + ∂ n ϕ ∂ τ ( ( cosh ⁡ τ − cos ⁡ σ ) ( 1 − cosh ⁡ τ cos ⁡ σ ) sinh ⁡ τ ) + ⋯ + ∂ n ϕ ∂ σ ( − ( cosh ⁡ τ − cos ⁡ σ ) sin ⁡ σ ) + ∂ 2 n ϕ ∂ τ 2 ( cosh ⁡ τ − cos ⁡ σ ) 2 + ⋯ + ∂ 2 n ϕ ∂ σ 2 ( cosh ⁡ τ − cos ⁡ σ ) 2 + ∂ 2 n ϕ ∂ ϕ 2 ( ( cosh ⁡ τ − cos ⁡ σ ) 2 sinh 2 ⁡ τ ) } {\displaystyle {\begin{aligned}\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{aligned}}}

Other differential operators such as ∇ ⋅ F {\displaystyle \nabla \cdot \mathbf {F} } and ∇ × F {\displaystyle \nabla \times \mathbf {F} } can be expressed in the coordinates ( σ , τ , ϕ ) {\displaystyle (\sigma ,\tau ,\phi )} by substituting the scale factors into the general formulae found in [orthogonal coordinates](/source/Orthogonal_coordinates).

## Toroidal harmonics

### Standard separation

The 3-variable [Laplace equation](/source/Laplace_equation)

- ∇ 2 Φ = 0 {\displaystyle \nabla ^{2}\Phi =0}

admits solution via [separation of variables](/source/Separation_of_variables) in toroidal coordinates. Making the substitution

- Φ = U cosh ⁡ τ − cos ⁡ σ {\displaystyle \Phi =U{\sqrt {\cosh \tau -\cos \sigma }}}

A separable equation is then obtained. A particular solution obtained by [separation of variables](/source/Separation_of_variables) is:

- Φ = cosh ⁡ τ − cos ⁡ σ S ν ( σ ) T μ ν ( τ ) V μ ( ϕ ) {\displaystyle \Phi ={\sqrt {\cosh \tau -\cos \sigma }}\,\,S_{\nu }(\sigma )T_{\mu \nu }(\tau )V_{\mu }(\phi )}

where each function is a [linear combination](/source/Linear_combination) of:

- S ν ( σ ) = e i ν σ a n d e − i ν σ {\displaystyle S_{\nu }(\sigma )=e^{i\nu \sigma }\,\,\,\,\mathrm {and} \,\,\,\,e^{-i\nu \sigma }}

- T μ ν ( τ ) = P ν − 1 / 2 μ ( cosh ⁡ τ ) a n d Q ν − 1 / 2 μ ( cosh ⁡ τ ) {\displaystyle T_{\mu \nu }(\tau )=P_{\nu -1/2}^{\mu }(\cosh \tau )\,\,\,\,\mathrm {and} \,\,\,\,Q_{\nu -1/2}^{\mu }(\cosh \tau )}

- V μ ( ϕ ) = e i μ ϕ a n d e − i μ ϕ {\displaystyle V_{\mu }(\phi )=e^{i\mu \phi }\,\,\,\,\mathrm {and} \,\,\,\,e^{-i\mu \phi }}

Where P and Q are [associated Legendre functions](/source/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 = cosh ⁡ τ > 1 {\displaystyle z=\cosh \tau >1} then, for instance, with vanishing order μ = 0 {\displaystyle \mu =0} (the convention is to not write the order when it vanishes) and ν = 0 {\displaystyle \nu =0}

- Q − 1 2 ( z ) = 2 1 + z K ( 2 1 + z ) {\displaystyle Q_{-{\frac {1}{2}}}(z)={\sqrt {\frac {2}{1+z}}}K\left({\sqrt {\frac {2}{1+z}}}\right)}

and

- P − 1 2 ( z ) = 2 π 2 1 + z K ( z − 1 z + 1 ) {\displaystyle P_{-{\frac {1}{2}}}(z)={\frac {2}{\pi }}{\sqrt {\frac {2}{1+z}}}K\left({\sqrt {\frac {z-1}{z+1}}}\right)}

where K {\displaystyle \,\!K} and E {\displaystyle \,\!E} are the complete [elliptic integrals](/source/Elliptic_integrals) of the [first](/source/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind) and [second](/source/Elliptic_integral#Complete_elliptic_integral_of_the_second_kind) 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](/source/Partial_differential_equations), e.g., [Laplace's equation](/source/Laplace's_equation) for which toroidal coordinates allow a [separation of variables](/source/Separation_of_variables) or the [Helmholtz equation](/source/Helmholtz_equation), for which toroidal coordinates do not allow a separation of variables. Typical examples would be the [electric potential](/source/Electric_potential) and [electric field](/source/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)

- Φ = U ρ {\displaystyle \Phi ={\frac {U}{\sqrt {\rho }}}}

where

- ρ = x 2 + y 2 = a sinh ⁡ τ cosh ⁡ τ − cos ⁡ σ . {\displaystyle \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](/source/Separation_of_variables) is then:

- Φ = a ρ S ν ( σ ) T μ ν ( τ ) V μ ( ϕ ) {\displaystyle \Phi ={\frac {a}{\sqrt {\rho }}}\,\,S_{\nu }(\sigma )T_{\mu \nu }(\tau )V_{\mu }(\phi )}

where each function is a linear combination of:

- S ν ( σ ) = e i ν σ a n d e − i ν σ {\displaystyle S_{\nu }(\sigma )=e^{i\nu \sigma }\,\,\,\,\mathrm {and} \,\,\,\,e^{-i\nu \sigma }}

- T μ ν ( τ ) = P μ − 1 / 2 ν ( coth ⁡ τ ) a n d Q μ − 1 / 2 ν ( coth ⁡ τ ) {\displaystyle T_{\mu \nu }(\tau )=P_{\mu -1/2}^{\nu }(\coth \tau )\,\,\,\,\mathrm {and} \,\,\,\,Q_{\mu -1/2}^{\nu }(\coth \tau )}

- V μ ( ϕ ) = e i μ ϕ a n d e − i μ ϕ . {\displaystyle 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 coth ⁡ τ {\displaystyle \coth \tau } rather than cosh ⁡ τ {\displaystyle \cosh \tau } and the μ {\displaystyle \mu } and ν {\displaystyle \nu } indices are exchanged. This method is useful for situations in which the boundary conditions are independent of the spherical angle θ {\displaystyle \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](/source/Whipple_formulae).

## References

- Byerly, W E. (1893) *[An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics](https://archive.org/details/elemtreatfour00byerrich)* Ginn & co. pp. 264–266

- Arfken G (1970). *Mathematical Methods for Physicists* (2nd ed.). Orlando, FL: Academic Press. pp. 112–115.

- Andrews, Mark (2006). "Alternative separation of Laplace's equation in toroidal coordinates and its application to electrostatics". *Journal of Electrostatics*. **64** (10): 664–672. [CiteSeerX](/source/CiteSeerX_(identifier)) [10.1.1.205.5658](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.205.5658). [doi](/source/Doi_(identifier)):[10.1016/j.elstat.2005.11.005](https://doi.org/10.1016%2Fj.elstat.2005.11.005).

- Hulme, A. (1982). "A note on the magnetic scalar potential of an electric current-ring". *[Mathematical Proceedings of the Cambridge Philosophical Society](/source/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society)*. **92** (1): 183–191. [Bibcode](/source/Bibcode_(identifier)):[1982MPCPS..92..183H](https://ui.adsabs.harvard.edu/abs/1982MPCPS..92..183H). [doi](/source/Doi_(identifier)):[10.1017/S0305004100059831](https://doi.org/10.1017%2FS0305004100059831).

## Bibliography

- Morse P M, Feshbach H (1953). *Methods of Theoretical Physics, Part I*. New York: McGraw–Hill. p. 666.

- Korn G A, [Korn T M](/source/Theresa_M._Korn) (1961). *Mathematical Handbook for Scientists and Engineers*. New York: McGraw-Hill. p. 182. [LCCN](/source/LCCN_(identifier)) [59014456](https://lccn.loc.gov/59014456).

- Margenau H, Murphy G M (1956). [*The Mathematics of Physics and Chemistry*](https://archive.org/details/mathematicsphysi00marg_501). New York: D. van Nostrand. pp. [190](https://archive.org/details/mathematicsphysi00marg_501/page/n203)–192. [LCCN](/source/LCCN_(identifier)) [55010911](https://lccn.loc.gov/55010911).

- Moon P H, Spencer D E (1988). "Toroidal Coordinates (*η*, *θ*, *ψ*)". *Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions* (2nd ed., 3rd revised printing ed.). New York: Springer Verlag. pp. 112–115 (Section IV, E4Ry). [ISBN](/source/ISBN_(identifier)) [978-0-387-02732-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-02732-6).

## External links

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

v t e Orthogonal coordinate systems Two dimensional Cartesian Polar (Log-polar) Parabolic Bipolar Elliptic Three dimensional Cartesian Cylindrical Spherical Parabolic Paraboloidal Oblate spheroidal Prolate spheroidal Ellipsoidal Elliptic cylindrical Toroidal Bispherical Bipolar cylindrical Conical 6-sphere

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