Toroidal coordinates

The focal ring is also known as the reference circle.

coordinate equals the natural logarithm of the ratio of the distances

to opposite sides of the focal ring The coordinate ranges are

correspond to spheres of different radii that all pass through the focal ring but are not concentric.

are non-intersecting tori of different radii that surround the focal ring.

of the point P is given by and its distances to the foci in the plane defined by

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 The scale factors for the toroidal coordinates

cosh ⁡ τ − cos ⁡ σ

cosh ⁡ τ − cos ⁡ σ

cosh ⁡ τ − cos ⁡ σ

( − sinh ⁡ τ sin ⁡ σ ) +

( cosh ⁡ τ − cos ⁡ σ ) ( 1 − cosh ⁡ τ cos ⁡ σ )

− 2 ( cosh ⁡ τ − cos ⁡ σ ) ( 1 − cosh ⁡ τ cos ⁡ σ )

( cosh ⁡ τ − cos ⁡ σ

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

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.

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

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

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

For identities relating the toroidal harmonics with argument hyperbolic cosine with those of argument hyperbolic cotangent, see the Whipple formulae.

Illustration of toroidal coordinates, which are obtained by rotating a two-dimensional bipolar coordinate system 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 $xy$-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).
Rotating this two-dimensional bipolar coordinate system 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 .
Geometric interpretation of the coordinates σ and τ of a point P . Observed in the plane of constant azimuthal angle , toroidal coordinates are equivalent to bipolar coordinates . 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 (magenta box); they are orthogonal.