Bipolar cylindrical coordinates

Bipolar cylindrical coordinates

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

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 (sigma, au, z) is

:x = a frac{sinh au}{cosh au - cos sigma}

:y = a frac{sin sigma}{cosh au - cos sigma}

:z = z

where the sigma coordinate of a point Pequals the angle F_{1} P F_{2} and the au coordinate equals the natural logarithm of the ratio of the distances d_{1} and d_{2} to the focal lines

: au = ln frac{d_{1{d_{2

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

Surfaces of constant sigma correspond to cylinders of different radii

:x^{2} +left( y - a cot sigma ight)^{2} = frac{a^{2{sin^{2} sigma}

that all pass through the focal lines and are not concentric. The surfaces of constant au are non-intersecting cylinders of different radii

:y^{2} +left( x - a coth au ight)^{2} = frac{a^{2{sinh^{2} au}

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

cale factors

The scale factors for the bipolar coordinates sigma and au are equal

:h_{sigma} = h_{ au} = frac{a}{cosh au - cossigma}

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

:dV = frac{a^{2{left( cosh au - cossigma ight)^{2 dsigma d au dz

and the Laplacian is given by

: abla^{2} Phi =frac{1}{a^{2 left( cosh au - cossigma ight)^{2}left( frac{partial^{2} Phi}{partial sigma^{2 + frac{partial^{2} Phi}{partial au^{2 ight) + frac{partial^{2} Phi}{partial z^{2

Other differential operators such as abla cdot mathbf{F} and abla imes mathbf{F} can be expressed in the coordinates (sigma, au) 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.

See also

Bibliography

* | pages = pp. 187–190

*, ASIN B0000CKZX7 | pages = p. 182

*

External links

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


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Bipolar coordinates — are a two dimensional orthogonal coordinate system. There are two commonly defined types of bipolar coordinates. [http://bbs.sachina.pku.edu.cn/Stat/Math World/math/b/b233.htm Eric W. Weisstein, Concise Encyclopedia of Mathematics CD ROM, Bipolar …   Wikipedia

  • Cylindrical coordinate system — A cylindrical coordinate system with origin O, polar axis A, and longitudinal axis L. The dot is the point with radial distance ρ = 4, angular coordinate φ = 130°, and height z = 4. A cylindrical coordinate system is …   Wikipedia

  • Curvilinear coordinates — Curvilinear, affine, and Cartesian coordinates in two dimensional space Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian… …   Wikipedia

  • Orthogonal coordinates — In mathematics, orthogonal coordinates are defined as a set of d coordinates q = (q1, q2, ..., qd) in which the coordinate surfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate surface for a particular… …   Wikipedia

  • Oblate spheroidal coordinates — Figure 1: Coordinate isosurfaces for a point P (shown as a black sphere) in oblate spheroidal coordinates (μ, ν, φ). The z axis is vertical, and the foci are at ±2. The red oblate spheroid (flattened sphere) corresponds to μ=1, whereas the blue… …   Wikipedia

  • Paraboloidal coordinates — are a three dimensional orthogonal coordinate system (λ,μ,ν) that generalizes the two dimensional parabolic coordinate system. Similar to the related ellipsoidal coordinates, the paraboloidal coordinate system has orthogonal quadratic coordinate… …   Wikipedia

  • Ellipsoidal coordinates — are a three dimensional orthogonal coordinate system (λ,μ,ν) that generalizes the two dimensional elliptic coordinate system. Unlike most three dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces, the ellipsoidal… …   Wikipedia

  • Conical coordinates — Coordinate surfaces of the conical coordinates. The constants b and c were chosen as 1 and 2, respectively. The red sphere represents r=2, the blue elliptic cone aligned with the vertical z axis represents μ=cosh(1) and the yellow elliptic cone… …   Wikipedia

  • Toroidal coordinates — are a three dimensional orthogonal coordinate system that results from rotating the two dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci F {1} and F {2} in bipolar coordinates become a ring of… …   Wikipedia

  • Coordinate system — For geographical coordinates on Wikipedia, see Wikipedia:WikiProject Geographical coordinates. In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”