|
|
| Line 1: |
Line 1: |
| {{distinguish|bipolar coordinates}}
| | The writer is called Araceli Gulledge. I am a production and distribution officer. The thing I adore most flower arranging and now I have time to consider on new issues. Delaware has usually been my residing location and will never transfer.<br><br>Also visit my weblog - extended auto warranty ([http://Www.Gahaghan.com/UserProfile/tabid/42/userId/44115/Default.aspx mouse click the following internet site]) |
| | |
| [[Image:Two-centerBipolarCoordinateSystem.JPG|right|frame|Two-center bipolar coordinates.]]
| |
| | |
| In [[mathematics]], '''two-center bipolar coordinates''' is a coordinate system, based on two coordinates which give distances from two fixed centers, <math>c_1</math> and <math>c_2</math>.<ref name=bip>{{mathworld|urlname=BipolarCoordinates|title=Bipolar coordinates}}</ref> This system is very useful in some {{which|date=May 2013}} scientific applications.<ref>[http://www.physics.utah.edu/~rprice/AREA51DOCS/paperIIa.pdf R. Price, The Periodic Standing Wave Approximation: Adapted coordinates and spectral methods.]</ref><ref>[http://arxiv.org/abs/gr-qc/0502034v1 The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method.]</ref>
| |
| | |
| ==Cartesian coordinates==
| |
| [[Image:Polar to cartesian.svg|thumb|right|350px|Cartesian coordinates and polar coordinates.]]
| |
| The transformation to Cartesian coordinates <math>(x,\ y)</math> from two-center bipolar coordinates <math>(r_1,\ r_2)</math> is
| |
| :<math>
| |
| x = \frac{r_1^2-r_2^2}{4a}
| |
| </math>
| |
| | |
| :<math>
| |
| y = \pm \frac{1}{4a}\sqrt{16a^2r_1^2-(r_1^2-r_2^2+4a^2)^2}
| |
| </math>
| |
| where the centers of this coordinate system are at <math>(+a,\ 0)</math> and <math>(-a,\ 0)</math>.<ref name=bip />
| |
| | |
| ==Polar coordinates==
| |
| The transformation to polar coordinates from two-center bipolar coordinates is
| |
| :<math>
| |
| r = \sqrt{\frac{r_1^2+r_2^2-2a^2}{2}}
| |
| </math>
| |
| | |
| :<math>\theta = \arctan \left( \frac{\sqrt{8a^2(r_1^2+r_2^2 - 2a^2)-(r_1^2 - r_2^2)^2}}{r_1^2 - r_2^2}\right)\,\!</math>
| |
| where <math>2 a</math> is the distance between the poles (coordinate system centers).
| |
| | |
| ==See also==
| |
| *[[Biangular coordinates]]
| |
| | |
| ==References==
| |
| <references/>
| |
| | |
| {{Orthogonal coordinate systems}}
| |
| | |
| [[Category:Coordinate systems|Two-center bipolar coordinates]]
| |
| | |
| | |
| {{geometry-stub}}
| |
The writer is called Araceli Gulledge. I am a production and distribution officer. The thing I adore most flower arranging and now I have time to consider on new issues. Delaware has usually been my residing location and will never transfer.
Also visit my weblog - extended auto warranty (mouse click the following internet site)