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).
Toroidal coordinates are a three-dimensional orthogonalcoordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci and in bipolar coordinates become a ring of radius in the plane of the toroidal coordinate system; the -axis is the axis of rotation. The focal ring is also known as the reference circle.
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.
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.
The coordinates may be calculated from the Cartesian coordinates (x, y, z) 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
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.
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 (the convention is to not write the order when it vanishes) 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.
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.
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