See also: two-center bipolar coordinates.
Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles.[1] Confusingly, the same term is also sometimes used for two-center bipolar coordinates. There is also a third system, based on two poles (biangular coordinates).
The term "bipolar" is further used on occasion to describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term bipolar coordinates is reserved for the coordinates described here, and never used for systems associated with those other curves, such as elliptic coordinates.
The system is based on two foci F1 and F2. Referring to the figure at right, the σ-coordinate of a point P equals the angle F1 P F2, and the τ-coordinate equals the natural logarithm of the ratio of the distances d1 and d2:
\tau=ln
| d1 | |
| d2 |
.
If, in the Cartesian system, the foci are taken to lie at (−a, 0) and (a, 0), the coordinates of the point P are
x=a
| \sinh\tau | |
| \cosh\tau-\cos\sigma |
, y=a
| \sin\sigma | |
| \cosh\tau-\cos\sigma |
.
The coordinate τ ranges from
-infty
infty
The equations for x and y can be combined to give
x+iy=ai\cot\left(
| \sigma+i\tau | |
| 2 |
\right)
x+iy=a\coth\left(
| \tau-i\sigma | |
| 2 |
\right).
This equation shows that σ and τ are the real and imaginary parts of an analytic function of x+iy (with logarithmic branch points at the foci), which in turn proves (by appeal to the general theory of conformal mapping) (the Cauchy-Riemann equations) that these particular curves of σ and τ intersect at right angles, i.e., it is an orthogonal coordinate system.
The curves of constant σ correspond to non-concentric circles
that intersect at the two foci. The centers of the constant-σ circles lie on the y-axis at
a\cot\sigma
\tfrac{a}{\sin\sigma}
The curves of constant
\tau
that surround the foci but again are not concentric. The centers of the constant-τ circles lie on the x-axis at
a\coth\tau
\tfrac{a}{\sinh\tau}
The passage from the Cartesian coordinates towards the bipolar coordinates can be done via the following formulas:
\tau=
| 1 | |
| 2 |
ln
| (x+a)2+y2 | |
| (x-a)2+y2 |
\pi-\sigma=2\arctan
| 2ay | |
| a2-x2-y2+\sqrt{(a2-x2-y2)2+4a2y2 |
}.
The coordinates also have the identities:
\tanh\tau=
| 2ax | |
| x2+y2+a2 |
\tan\sigma=
| 2ay | |
| x2+y2-a2 |
,
\cot\sigma
\coth\tau
To obtain the scale factors for bipolar coordinates, we take the differential of the equation for
x+iy
dx+idy=
| -ia | |
| \sin2l(\tfrac{1 |
{2}(\sigma+i\tau)r)}(d\sigma+id\tau).
(dx)2+(dy)2=
| a2 | |
| l[2\sin\tfrac{1 |
{2}l(\sigma+i\taur)\sin\tfrac{1}{2}l(\sigma-i\taur)r]2}l((d\sigma)2+(d\tau)2r).
2\sin\tfrac{1}{2}l(\sigma+i\taur)\sin\tfrac{1}{2}l(\sigma-i\taur) =\cos\sigma-\cosh\tau,
(dx)2+(dy)2=
| a2 | |
| (\cosh\tau-\cos\sigma)2 |
l((d\sigma)2+(d\tau)2r).
Hence the scale factors for σ and τ are equal, and given by
h\sigma=h\tau=
| a | |
| \cosh\tau-\cos\sigma |
.
Many results now follow in quick succession from the general formulae for orthogonal coordinates.Thus, the infinitesimal area element equals
dA=
| a2 | |
| \left(\cosh\tau-\cos\sigma\right)2 |
d\sigmad\tau,
and the Laplacian is given by
\nabla2\Phi=
| 1 | |
| a2 |
\left(\cosh\tau-\cos\sigma\right)2 \left(
| \partial2\Phi | |
| \partial\sigma2 |
+
| \partial2\Phi | |
| \partial\tau2 |
\right).
Expressions for
\nablaf
\nabla ⋅ F
\nabla x F
The classic applications of bipolar coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which bipolar coordinates allow a separation of variables. An example is the electric field surrounding two parallel cylindrical conductors with unequal diameters.
Bipolar coordinates form the basis for several sets of three-dimensional orthogonal coordinates.