The most common definition of toroidal coordinates is together with ). The coordinate of a point equals the angle and the coordinate equals the natural logarithm of the ratio of the distances and to opposite sides of the focal ring The coordinate ranges are and and
Coordinate surfaces
Surfaces of constant correspond to spheres of different radii that all pass through the focal ring but are not concentric. The surfaces of constant are non-intersecting tori of different radii that surround the focal ring. The centers of the constant- spheres lie along the -axis, whereas the constant- tori are centered in the plane.
Inverse transformation
The coordinates may be calculated from the Cartesian coordinates as follows. The azimuthal angle is given by the formula The cylindrical radius of the point P is given by and its distances to the foci in the plane defined by is given by . The angle is formed by the two foci in this plane and P, whereas is the logarithm of the ratio of distances to the foci. The corresponding circles of constant and are shown in red and blue, respectively, and meet at right angles ; they are orthogonal. The coordinate equals the natural logarithm of the focal distances whereas equals the angle between the rays to the foci, which may be determined from the law of cosines Or explicitly, including the sign, where. The transformations between cylindrical and toroidal coordinates can be expressed in complex notation as
Scale factors
The scale factors for the toroidal coordinates and are equal whereas the azimuthal scale factor equals Thus, the infinitesimal volume element equals and the Laplacian is given by Other differential operators such as and can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates.
Toroidal harmonics
Standard separation
The 3-variable Laplace equation admits solution via separation of variables in toroidal coordinates. Making the substitution A separable equation is then obtained. A particular solution obtained by separation of variables is: where each function is a linear combination of: 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 then, for instance, with vanishing order and and where and 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.
An alternative separation
Alternatively, a different substitution may be made where Again, a separable equation is obtained. A particular solution obtained by separation of variables is then: where each function is a linear combination of: Note that although the toroidal harmonics are used again for the T function, the argument is rather than and the and indices are exchanged. This method is useful for situations in which the boundary conditions are independent of the spherical angle, 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.
