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 F_{1} and F_{2} in bipolar coordinates remain points (on the z-axis, the axis of rotation) in the bispherical coordinate system.
Definition
The most common definition of bispherical coordinates (\tau, \sigma, \phi) is
\begin{align} x &= a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \cos \phi, \\ y &= a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \sin \phi, \\ z &= a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma}, \end{align}
where the \sigma coordinate of a point P equals the angle F_{1} P F_{2} and the \tau coordinate equals the natural logarithm of the ratio of the distances d_{1} and d_{2} to the foci
\tau = \ln \frac{d_{1}}{d_{2}}
The coordinates ranges are −∞ < \tau < ∞, 0 ≤ \sigma ≤ \pi and 0 ≤ \phi ≤ 2\pi.
Coordinate surfaces
Surfaces of constant \sigma correspond to intersecting tori of different radii
z^{2} + \left( \sqrt{x^2 + y^2} - a \cot \sigma \right)^2 = \frac{a^2}{\sin^2 \sigma}
that all pass through the foci but are not concentric. The surfaces of constant \tau are non-intersecting spheres of different radii
\left( x^2 + y^2 \right) + \left( z - a \coth \tau \right)^2 = \frac{a^2}{\sinh^2 \tau}
that surround the foci. The centers of the constant-\tau spheres lie along the z-axis, whereas the constant-\sigma tori are centered in the xy plane.
Inverse formulae
The formulae for the inverse transformation are:
\begin{align} \sigma &= \arccos\left(\dfrac{R^2-a^2}{Q}\right), \\ \tau &= \operatorname{arsinh}\left(\dfrac{2az}{Q}\right), \\ \phi &= \arctan\left(\dfrac{y}{x}\right), \end{align}
where R = \sqrt{x^2 + y^2 + z^2} and Q = \sqrt{\left(R^2 + a^2\right)^2 - \left(2 a z\right)^2}.
Scale factors
The scale factors for the bispherical coordinates \sigma and \tau are equal
h_\sigma = h_\tau = \frac{a}{\cosh \tau - \cos\sigma}
whereas the azimuthal scale factor equals
h_\phi = \frac{a \sin \sigma}{\cosh \tau - \cos\sigma}
Thus, the infinitesimal volume element equals
dV = \frac{a^3 \sin \sigma}{\left( \cosh \tau - \cos\sigma \right)^3} \, d\sigma \, d\tau \, d\phi
and the Laplacian is given by
\begin{align} \nabla^2 \Phi = \frac{\left( \cosh \tau - \cos\sigma \right)^3}{a^2 \sin \sigma} & \left[ \frac{\partial}{\partial \sigma} \left( \frac{\sin \sigma}{\cosh \tau - \cos\sigma} \frac{\partial \Phi}{\partial \sigma} \right) \right. \\[8pt] &{} \quad + \left. \sin \sigma \frac{\partial}{\partial \tau} \left( \frac{1}{\cosh \tau - \cos\sigma} \frac{\partial \Phi}{\partial \tau} \right) + \frac{1}{\sin \sigma \left( \cosh \tau - \cos\sigma \right)} \frac{\partial^2 \Phi}{\partial \phi^2} \right] \end{align}
Other differential operators such as \nabla \cdot \mathbf{F} and \nabla \times \mathbf{F} can be expressed in the coordinates (\sigma, \tau) by substituting the scale factors into the general formulae found in orthogonal coordinates.
Applications
The classic applications of bispherical coordinates are in solving partial differential equations, e.g., Laplace's equation, for which bispherical coordinates allow a separation of variables. However, the Helmholtz equation is not separable in bispherical coordinates. A typical example would be the electric field surrounding two conducting spheres of different radii.
Bibliography
- Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Parts I and II. New York: McGraw-Hill. pp. 665–666, 1298–1301.
- Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 182. LCCN 59014456.
- Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. p. 113. ISBN 0-86720-293-9.
- Moon PH, Spencer DE (1988). "Bispherical Coordinates (η, θ, ψ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions. corrected 2nd ed., 3rd print ed. New York: Springer Verlag. pp. 110–112 (Section IV, E4Rx). ISBN 0-387-02732-7.