F1
F2
a
xy
z
The most common definition of toroidal coordinates
(\tau,\sigma,\phi)
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 |
sign(\sigma)=sign(z
\sigma
P
F1PF2
\tau
d1
d2
\tau=ln
d1 | |
d2 |
.
The coordinate ranges are
-\pi<\sigma\le\pi
\tau\ge0
0\le\phi<2\pi.
Surfaces of constant
\sigma
\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
z2+ \left(\sqrt{x2+y2
that surround the focal ring. The centers of the constant-
\sigma
z
\tau
xy
The
(\sigma,\tau,\phi)
\phi
\tan\phi=
y | |
x |
The cylindrical radius
\rho
\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
2 | |
d | |
1 |
=(\rho+a)2+z2
2 | |
d | |
2 |
=(\rho-a)2+z2
The coordinate
\tau
\tau=ln
d1 | |
d2 |
whereas
|\sigma|
\cos\sigma=
| |||||||||||||||
2d1d2 |
.
\sigma=sign(z)\arccos
r2-a2 | |
\sqrt{(r2-a2)2+4a2z2 |
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) |
.
The scale factors for the toroidal coordinates
\sigma
\tau
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
The Laplacian is given by
For a vector field the Vector Laplacian is given by
Other differential operators such as
\nabla ⋅ F
\nabla x F
(\sigma,\tau,\phi)
\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
\mu=0
\nu=0
Q | (z)=\sqrt{ | ||||
|
2 | |
1+z |
and
P | (z)= | ||||
|
2 | \sqrt{ | |
\pi |
2 | |
1+z |
where
K
E
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= | U |
\sqrt{\rho |
where
\rho=\sqrt{x2+y
| ||||
.
Again, a separable equation is obtained. A particular solution obtained by separation of variables is then:
\Phi=
a | |
\sqrt{\rho |
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
\cosh\tau
\mu
\nu
\theta