# Bicylindrical coordinates

The numbers $\tau , \sigma$ and $z$ related to the rectangular Cartesian coordinates $x, y$ and $z$ by the formulas

$$x = \frac{a \sinh \tau }{\cosh \tau - \cos \sigma } ,\ \ y = \frac{a \sin \sigma }{\cosh \tau - \cos \sigma } ,\ \ z = z,$$

where $0 \leq \sigma < \pi , -\infty < \tau < \infty$. The coordinate surfaces are: the family of pairs of circular cylinders with parallel axes ( $\tau = \textrm{ const }$), the family of circular cylinders orthogonal to the former ( $\sigma = \textrm{ const }$), and the planes ( $z = \textrm{ const }$). The system of bicylindrical coordinates is obtained as the result of translation of the system of bipolar coordinates in the $xy$- plane parallel to the $z$- axis.

The Lamé coefficients (cf. Lamé coefficients) are:

$$L _ \sigma = L _ \tau = \ \frac{a ^ {2} }{( \cosh \tau - \cos \sigma ) ^ {2} } ,\ \ L _ {z} = 1.$$

The Laplace operator is:

$$\Delta f = \frac{1}{a ^ {2} } ( \cosh \tau - \cos \sigma ) ^ {2} \left ( \frac{\partial ^ {2} f }{\partial \sigma ^ {2} } + \frac{\partial ^ {2} f }{\partial \tau ^ {2} } \ \right ) + \frac{\partial ^ {2} f }{\partial z ^ {2} } .$$

How to Cite This Entry:
Bicylindrical coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bicylindrical_coordinates&oldid=46051
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article