The term "bipolar" is often used to describe other curves having two singular points (foci), such as
ellipses,
hyperbolas, and
Cassini ovals. However, the term bipolar coordinates is never used to describe coordinates associated with those curves, e.g.,
elliptic coordinates.
Basic definition
The most common definition of bipolar cylindrical coordinates is
where the coordinate of a point
equals the angle and the
coordinate equals the
natural logarithm of the ratio of the distances and to the focal lines
(Recall that the focal lines and are located at and , respectively.)
Surfaces of constant correspond to cylinders of different radii
that all pass through the focal lines and are not concentric. The surfaces of constant are non-intersecting cylinders of different radii
that surround the focal lines but again are not concentric. The focal lines and all these cylinders are parallel to the -axis (the direction of projection). In the plane, the centers of the constant- and constant- cylinders lie on the and axes, respectively.
Scale factors
The scale factors for the bipolar coordinates and are equal
whereas the remaining scale factor .
Thus, the infinitesimal volume element equals
and the Laplacian is given by
Other differential operators such as
and can be expressed in the coordinates by substituting
the scale factors into the general formulae
found in
orthogonal coordinates.
Korn GA,
Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 182.
LCCN59014456. ASIN B0000CKZX7.
Moon P, Spencer DE (1988). "Conical Coordinates (r, θ, λ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. unknown.
ISBN978-0-387-18430-2.