Orthogonal coordinates

2013-12-20T18:12:09Z

31.185.211.85: /* Differential operators in three dimensions */ Add boldface on vector quantity

[[File:Bipolar cylindrical coordinates.png|thumb|350px|right|[[Coordinate system#Coordinate surface|Coordinate surfaces]] of the bipolar cylindrical coordinates. The yellow crescent corresponds to σ, whereas the red tube corresponds to τ and the blue plane corresponds to ''z''=1. The three surfaces intersect at the point '''P''' (shown as a black sphere).]]

'''Bipolar cylindrical coordinates''' are a three-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] that results from projecting the two-dimensional [[bipolar coordinates|bipolar coordinate system]] in the
perpendicular <math>z</math>-direction. The two lines of [[Focus (geometry)|foci]]
<math>F_{1}</math> and <math>F_{2}</math> of the projected [[Apollonian circles]] are generally taken to be
defined by <math>x=-a</math> and <math>x=+a</math>, respectively, (and by <math>y=0</math>) in the [[Cartesian coordinate system]].

The term "bipolar" is often used to describe other curves having two singular points (foci), such as [[ellipse]]s, [[hyperbola]]s, and [[Cassini oval]]s. 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 <math>(\sigma, \tau, z)</math> is

:<math>
x = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma}
</math>

:<math>
y = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma}
</math>

:<math>
z = \ z
</math>

where the <math>\sigma</math> coordinate of a point <math>P</math>
equals the angle <math>F_{1} P F_{2}</math> and the
<math>\tau</math> coordinate equals the [[natural logarithm]] of the ratio of the distances <math>d_{1}</math> and <math>d_{2}</math> to the focal lines

:<math>
\tau = \ln \frac{d_{1}}{d_{2}}
</math>

(Recall that the focal lines <math>F_{1}</math> and <math>F_{2}</math> are located at <math>x=-a</math> and <math>x=+a</math>, respectively.)

Surfaces of constant <math>\sigma</math> correspond to cylinders of different radii

:<math>
x^{2} +
\left( y - a \cot \sigma \right)^{2} = \frac{a^{2}}{\sin^{2} \sigma}
</math>

that all pass through the focal lines and are not concentric. The surfaces of constant <math>\tau</math> are non-intersecting cylinders of different radii

:<math>
y^{2} +
\left( x - a \coth \tau \right)^{2} = \frac{a^{2}}{\sinh^{2} \tau}
</math>

that surround the focal lines but again are not concentric. The focal lines and all these cylinders are parallel to the <math>z</math>-axis (the direction of projection). In the <math>z=0</math> plane, the centers of the constant-<math>\sigma</math> and constant-<math>\tau</math> cylinders lie on the <math>y</math> and <math>x</math> axes, respectively.

==Scale factors==

The scale factors for the bipolar coordinates <math>\sigma</math> and <math>\tau</math> are equal

:<math>
h_{\sigma} = h_{\tau} = \frac{a}{\cosh \tau - \cos\sigma}
</math>

whereas the remaining scale factor <math>h_{z}=1</math>.
Thus, the infinitesimal volume element equals

:<math>
dV = \frac{a^{2}}{\left( \cosh \tau - \cos\sigma \right)^{2}} d\sigma d\tau dz
</math>

and the Laplacian is given by

:<math>
\nabla^{2} \Phi =
\frac{1}{a^{2}} \left( \cosh \tau - \cos\sigma \right)^{2}
\left(
\frac{\partial^{2} \Phi}{\partial \sigma^{2}} +
\frac{\partial^{2} \Phi}{\partial \tau^{2}}
\right) +
\frac{\partial^{2} \Phi}{\partial z^{2}}
</math>

Other differential operators such as <math>\nabla \cdot \mathbf{F}</math>
and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\sigma, \tau)</math> by substituting
the scale factors into the general formulae
found in [[orthogonal coordinates]].

==Applications==
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]]. A typical example would be the [[electric field]] surrounding two
parallel cylindrical conductors.

==Bibliography==
*{{cite book | author = [[Henry Margenau|Margenau H]], Murphy GM | year = 1956 | title = The Mathematics of Physics and Chemistry | publisher = D. van Nostrand | location = New York | pages = 187–190 | lccn = 5510911 }}
*{{cite book | author = Korn GA, Korn TM |year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | id = ASIN B0000CKZX7 | page = 182 | lccn = 5914456}}
*{{cite book | author = Moon P, Spencer DE | year = 1988 | chapter = Conical Coordinates (r, θ, λ) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = corrected 2nd ed., 3rd print | publisher = Springer-Verlag | location = New York | isbn = 978-0-387-18430-2 | nopp = true | page = unknown}}

==External links==
*[http://mathworld.wolfram.com/BipolarCylindricalCoordinates.html MathWorld description of bipolar cylindrical coordinates]

{{Orthogonal coordinate systems}}

[[Category:Coordinate systems]]

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Orthogonal coordinates