Bipolar cylindrical coordinates are a three-dimensional
orthogonal coordinate system that results from projecting the two-dimensional
bipolar coordinate system in the
perpendicular
-direction. The two lines of
foci
Focus, or its plural form foci may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film
*''Focus'', a 1962 TV film starring James Whitmore
* ''Focus'' (2001 film), a 2001 film based ...
and
of the projected
Apollonian circles
In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. Th ...
are generally taken to be
defined by
and
, respectively, (and by
) in the
Cartesian coordinate system.
The term "bipolar" is often used to describe other curves having two singular points (foci), such as
ellipses,
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
s, and
Cassini oval
In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points ( foci) is constant. This may be contrasted with an ellipse, for which the ''sum'' of t ...
s. However, the term ''bipolar coordinates'' is never used to describe coordinates associated with those curves, e.g.,
elliptic coordinates
In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci F_ and F_ are generally taken to be fixed at -a and +a, respectively ...
.
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 In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
.
Applications
The classic applications of bipolar coordinates are in solving
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
,
e.g.,
Laplace's equation or the
Helmholtz equation, for which bipolar coordinates allow a
separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
(in 2D). A typical example would be the
electric field surrounding two
parallel cylindrical conductors.
Bibliography
*
*
*
External links
MathWorld description of bipolar cylindrical coordinates
{{Orthogonal coordinate systems
Three-dimensional coordinate systems
Orthogonal coordinate systems