Toroidal coordinates

Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci ${\displaystyle F_{1}}$ and ${\displaystyle F_{2}}$ in bipolar coordinates become a ring of radius ${\displaystyle a}$ in the ${\displaystyle xy}$ plane of the toroidal coordinate system; the ${\displaystyle z}$-axis is the axis of rotation. The focal ring is also known as the reference circle.

The most common definition of toroidal coordinates ${\displaystyle (\tau ,\sigma ,\phi )}$ is

together with ${\displaystyle \mathrm {sign} (\sigma )=\mathrm {sign} (z}$). The ${\displaystyle \sigma }$ coordinate of a point ${\displaystyle P}$ equals the angle ${\displaystyle F_{1}PF_{2}}$ and the ${\displaystyle \tau }$ coordinate equals the natural logarithm of the ratio of the distances ${\displaystyle d_{1}}$ and ${\displaystyle d_{2}}$ to opposite sides of the focal ring

The coordinate ranges are ${\displaystyle -\pi <\sigma \leq \pi }$, ${\displaystyle \tau \geq 0}$ and ${\displaystyle 0\leq \phi <2\pi .}$

Surfaces of constant ${\displaystyle \sigma }$ correspond to spheres of different radii

that all pass through the focal ring but are not concentric. The surfaces of constant ${\displaystyle \tau }$ are non-intersecting tori of different radii

that surround the focal ring. The centers of the constant-${\displaystyle \sigma }$ spheres lie along the ${\displaystyle z}$-axis, whereas the constant-${\displaystyle \tau }$ tori are centered in the ${\displaystyle xy}$ plane.

The ${\displaystyle (\sigma ,\tau ,\phi )}$ coordinates may be calculated from the Cartesian coordinates (x, y, z) as follows. The azimuthal angle ${\displaystyle \phi }$ is given by the formula