Bispherical coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that connects the two foci. Thus, the two foci ${\displaystyle F_{1}}$ and ${\displaystyle F_{2}}$ in bipolar coordinates remain points (on the ${\displaystyle z}$-axis, the axis of rotation) in the bispherical coordinate system.

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

where 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 the foci

The coordinates ranges are −∞ < ${\displaystyle \tau }$ < ∞, 0 ≤ ${\displaystyle \sigma }$ ≤ ${\displaystyle \pi }$ and 0 ≤ ${\displaystyle \phi }$ ≤ 2${\displaystyle \pi }$.

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

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

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

The formulae for the inverse transformation are: