- Source: 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
c
1
{\displaystyle c_{1}}
and
c
2
{\displaystyle c_{2}}
. This system is very useful in some scientific applications (e.g. calculating the electric field of a dipole on a plane).
Transformation to Cartesian coordinates
When the centers are at
(
+
a
,
0
)
{\displaystyle (+a,0)}
and
(
−
a
,
0
)
{\displaystyle (-a,0)}
, the transformation to Cartesian coordinates
(
x
,
y
)
{\displaystyle (x,y)}
from two-center bipolar coordinates
(
r
1
,
r
2
)
{\displaystyle (r_{1},r_{2})}
is
x
=
r
2
2
−
r
1
2
4
a
{\displaystyle x={\frac {r_{2}^{2}-r_{1}^{2}}{4a}}}
y
=
±
1
4
a
16
a
2
r
2
2
−
(
r
2
2
−
r
1
2
+
4
a
2
)
2
{\displaystyle y=\pm {\frac {1}{4a}}{\sqrt {16a^{2}r_{2}^{2}-(r_{2}^{2}-r_{1}^{2}+4a^{2})^{2}}}}
Transformation to polar coordinates
When x > 0, the transformation to polar coordinates from two-center bipolar coordinates is
r
=
r
1
2
+
r
2
2
−
2
a
2
2
{\displaystyle r={\sqrt {\frac {r_{1}^{2}+r_{2}^{2}-2a^{2}}{2}}}}
θ
=
arctan
(
r
1
4
−
8
a
2
r
1
2
−
2
r
1
2
r
2
2
−
(
4
a
2
−
r
2
2
)
2
r
2
2
−
r
1
2
)
{\displaystyle \theta =\arctan \left({\frac {\sqrt {r_{1}^{4}-8a^{2}r_{1}^{2}-2r_{1}^{2}r_{2}^{2}-(4a^{2}-r_{2}^{2})^{2}}}{r_{2}^{2}-r_{1}^{2}}}\right)}
where
2
a
{\displaystyle 2a}
is the distance between the poles (coordinate system centers).
Applications
Polar plotters use two-center bipolar coordinates to describe the drawing paths required to draw a target image.
See also
References
Kata Kunci Pencarian:
- Daftar transformasi koordinat
- Two-center bipolar coordinates
- Bipolar coordinates
- Biangular coordinates
- Polar plotter
- Bipolar cylindrical coordinates
- Lemniscate of Bernoulli
- Toroidal coordinates
- Cassini oval
- Cartesian oval
- Bispherical coordinates
