Toroidal coordinates explained

F1

and

F2

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.

Definition

The most common definition of toroidal coordinates

(\tau,\sigma,\phi)

is

x=a

\sinh\tau
\cosh\tau-\cos\sigma

\cos\phi

y=a

\sinh\tau
\cosh\tau-\cos\sigma

\sin\phi

z=a

\sin\sigma
\cosh\tau-\cos\sigma

together with

sign(\sigma)=sign(z

).The

\sigma

coordinate of a point

P

equals the angle

F1PF2

and the

\tau

coordinate equals the natural logarithm of the ratio of the distances

d1

and

d2

to opposite sides of the focal ring

\tau=ln

d1
d2

.

The coordinate ranges are

-\pi<\sigma\le\pi

,

\tau\ge0

and

0\le\phi<2\pi.

Coordinate surfaces

Surfaces of constant

\sigma

correspond to spheres of different radii

\left(x2+y2\right)+ \left(z-a\cot\sigma\right)2=

a2
\sin2\sigma

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

\tau

are non-intersecting tori of different radii

z2+ \left(\sqrt{x2+y2

} - a \coth \tau \right)^ = \frac

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

\sigma

spheres lie along the

z

-axis, whereas the constant-

\tau

tori are centered in the

xy

plane.

Inverse transformation

The

(\sigma,\tau,\phi)

coordinates may be calculated from the Cartesian coordinates (x, y, z) as follows. The azimuthal angle

\phi

is given by the formula

\tan\phi=

y
x

The cylindrical radius

\rho

of the point P is given by

\rho2=x2+y2=\left(a

\sinh\tau
\cosh\tau-\cos\sigma

\right)2

and its distances to the foci in the plane defined by

\phi

is given by
2
d
1

=(\rho+a)2+z2

2
d
2

=(\rho-a)2+z2

The coordinate

\tau

equals the natural logarithm of the focal distances

\tau=ln

d1
d2

whereas

|\sigma|

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

\cos\sigma=

2
d
1
+
2
d
2
-4a2
2d1d2

.

Or explicitly, including the sign,

\sigma=sign(z)\arccos

r2-a2
\sqrt{(r2-a2)2+4a2z2
}where

r=\sqrt{\rho2+z2}

.

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

z+i\rho =ia\coth

\tau+i\sigma
2

,

\tau+i\sigma =ln

z+i(\rho+a)
z+i(\rho-a)

.

Scale factors

The scale factors for the toroidal coordinates

\sigma

and

\tau

are equal

h\sigma=h\tau=

a
\cosh\tau-\cos\sigma

whereas the azimuthal scale factor equals

h\phi=

a\sinh\tau
\cosh\tau-\cos\sigma

Thus, the infinitesimal volume element equals

dV=

a3\sinh\tau
\left(\cosh\tau-\cos\sigma\right)3

d\sigmad\taud\phi

Differential Operators

The Laplacian is given by \begin\nabla^2 \Phi =\frac & \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\left(\frac\frac\right) + \frac\frac\right]\end

For a vector field \vec(\tau,\sigma,\phi) = n_(\tau,\sigma,\phi)\hat_ + n_(\tau,\sigma,\phi) \hat_ + n_ (\tau,\sigma,\phi) \hat_, the Vector Laplacian is given by\begin\Delta \vec(\tau,\sigma,\phi) &= \nabla (\nabla \cdot \vec) - \nabla \times (\nabla \times \vec) \\&= \frac\vec_ \left \\\&+ \frac\vec_ \left \\\&+ \frac\vec_ \left \\end

Other differential operators such as

\nablaF

and

\nabla x F

can be expressed in the coordinates

(\sigma,\tau,\phi)

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

Toroidal harmonics

Standard separation

\nabla2\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:

\Phi=\sqrt{\cosh\tau-\cos\sigma}S\nu(\sigma)T\mu\nu(\tau)V\mu(\phi)

where each function is a linear combination of:

i\nu\sigma
S
\nu(\sigma)=e

ande-i\nu\sigma

T\mu\nu

\mu(\cosh\tau)
(\tau)=P
\nu-1/2
i\mu\phi
V
\mu(\phi)=e

ande-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=\cosh\tau>1

then, for instance, with vanishing order

\mu=0

(the convention is to not write the order when it vanishes) and

\nu=0

Q(z)=\sqrt{
-12
2
1+z
}K\left(\sqrt\right)

and

P(z)=
-12
2\sqrt{
\pi
2
1+z
}K \left(\sqrt \right)

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 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)

\Phi=U
\sqrt{\rho
}

where

\rho=\sqrt{x2+y

2}=a\sinh\tau
\cosh\tau-\cos\sigma

.

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

\Phi=

a
\sqrt{\rho
}\,\,S_\nu(\sigma)T_(\tau)V_\mu(\phi)

where each function is a linear combination of:

i\nu\sigma
S
\nu(\sigma)=e

ande-i\nu\sigma

T\mu\nu

\nu(\coth\tau)
(\tau)=P
\mu-1/2
i\mu\phi
V
\mu(\phi)=e

ande-i\mu\phi.

Note that although the toroidal harmonics are used again for the T  function, the argument is

\coth\tau

rather than

\cosh\tau

and the

\mu

and

\nu

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

\theta

, 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.

References

Bibliography

External links