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( T o p )
1
D e f i n i t i o n
T o g g l e D e f i n i t i o n s u b s e c t i o n
1 . 1
C o o r d i n a t e s u r f a c e s
1 . 2
I n v e r s e f o r m u l a e
1 . 3
S c a l e f a c t o r s
2
A p p l i c a t i o n s
3
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4
B i b l i o g r a p h y
5
E x t e r n a l l i n k s
T o g g l e t h e t a b l e o f c o n t e n t s
B i s p h e r i c a l c o o r d i n a t e s
4 l a n g u a g e s
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P r i n t / e x p o r t
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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Illustration of bispherical coordinates, which are obtained by rotating a two-dimensional bipolar coordinate system about the axis joining its two foci. The foci are located at distance 1 from the vertical z -axis. The red self-intersecting torus is the σ=45° isosurface, the blue sphere is the τ=0.5 isosurface, and the yellow half-plane is the φ=60° isosurface. The green half-plane marks the x -z plane, from which φ is measured. The black point is located at the intersection of the red, blue and yellow isosurfaces, at Cartesian coordinates roughly (0.841, -1.456, 1.239).
Bispherical coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that connects the two foci. Thus, the two foci
F
1
{\displaystyle F_{1}}
and
F
2
{\displaystyle F_{2}}
in bipolar coordinates remain points (on the
z
{\displaystyle z}
-axis, the axis of rotation) in the bispherical coordinate system.
Definition
[ edit ]
The most common definition of bispherical coordinates
(
τ
,
σ
,
ϕ
)
{\displaystyle (\tau ,\sigma ,\phi )}
is
x
=
a
sin
σ
cosh
τ
−
cos
σ
cos
ϕ
,
y
=
a
sin
σ
cosh
τ
−
cos
σ
sin
ϕ
,
z
=
a
sinh
τ
cosh
τ
−
cos
σ
,
{\displaystyle {\begin{aligned}x&=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}\cos \phi ,\\y&=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}\sin \phi ,\\z&=a\ {\frac {\sinh \tau }{\cosh \tau -\cos \sigma }},\end{aligned}}}
where the
σ
{\displaystyle \sigma }
coordinate of a point
P
{\displaystyle P}
equals the angle
F
1
P
F
2
{\displaystyle F_{1}PF_{2}}
and the
τ
{\displaystyle \tau }
coordinate equals the natural logarithm of the ratio of the distances
d
1
{\displaystyle d_{1}}
and
d
2
{\displaystyle d_{2}}
to the foci
τ
=
ln
d
1
d
2
{\displaystyle \tau =\ln {\frac {d_{1}}{d_{2}}}}
The coordinates ranges are -∞ <
τ
{\displaystyle \tau }
< ∞, 0 ≤
σ
{\displaystyle \sigma }
≤
π
{\displaystyle \pi }
and 0 ≤
ϕ
{\displaystyle \phi }
≤ 2
π
{\displaystyle \pi }
.
Coordinate surfaces
[ edit ]
Surfaces of constant
σ
{\displaystyle \sigma }
correspond to intersecting tori of different radii
z
2
+
(
x
2
+
y
2
−
a
cot
σ
)
2
=
a
2
sin
2
σ
{\displaystyle z^{2}+\left({\sqrt {x^{2}+y^{2}}}-a\cot \sigma \right)^{2}={\frac {a^{2}}{\sin ^{2}\sigma }}}
that all pass through the foci but are not concentric. The surfaces of constant
τ
{\displaystyle \tau }
are non-intersecting spheres of different radii
(
x
2
+
y
2
)
+
(
z
−
a
coth
τ
)
2
=
a
2
sinh
2
τ
{\displaystyle \left(x^{2}+y^{2}\right)+\left(z-a\coth \tau \right)^{2}={\frac {a^{2}}{\sinh ^{2}\tau }}}
that surround the foci. The centers of the constant-
τ
{\displaystyle \tau }
spheres lie along the
z
{\displaystyle z}
-axis, whereas the constant-
σ
{\displaystyle \sigma }
tori are centered in the
x
y
{\displaystyle xy}
plane.
[ edit ]
The formulae for the inverse transformation are:
σ
=
arccos
(
R
2
−
a
2
Q
)
,
τ
=
arsinh
(
2
a
z
Q
)
,
ϕ
=
arctan
(
y
x
)
,
{\displaystyle {\begin{aligned}\sigma &=\arccos \left({\dfrac {R^{2}-a^{2}}{Q}}\right),\\\tau &=\operatorname {arsinh} \left({\dfrac {2az}{Q}}\right),\\\phi &=\arctan \left({\dfrac {y}{x}}\right),\end{aligned}}}
where
R
=
x
2
+
y
2
+
z
2
{\textstyle R={\sqrt {x^{2}+y^{2}+z^{2}}}}
and
Q
=
(
R
2
+
a
2
)
2
−
(
2
a
z
)
2
.
{\textstyle Q={\sqrt {\left(R^{2}+a^{2}\right)^{2}-\left(2az\right)^{2}}}.}
Scale factors
[ edit ]
The scale factors for the bispherical coordinates
σ
{\displaystyle \sigma }
and
τ
{\displaystyle \tau }
are equal
h
σ
=
h
τ
=
a
cosh
τ
−
cos
σ
{\displaystyle h_{\sigma }=h_{\tau }={\frac {a}{\cosh \tau -\cos \sigma }}}
whereas the azimuthal scale factor equals
h
ϕ
=
a
sin
σ
cosh
τ
−
cos
σ
{\displaystyle h_{\phi }={\frac {a\sin \sigma }{\cosh \tau -\cos \sigma }}}
Thus, the infinitesimal volume element equals
d
V
=
a
3
sin
σ
(
cosh
τ
−
cos
σ
)
3
d
σ
d
τ
d
ϕ
{\displaystyle dV={\frac {a^{3}\sin \sigma }{\left(\cosh \tau -\cos \sigma \right)^{3}}}\,d\sigma \,d\tau \,d\phi }
and the Laplacian is given by
∇
2
Φ
=
(
cosh
τ
−
cos
σ
)
3
a
2
sin
σ
[
∂
∂
σ
(
sin
σ
cosh
τ
−
cos
σ
∂
Φ
∂
σ
)
+
sin
σ
∂
∂
τ
(
1
cosh
τ
−
cos
σ
∂
Φ
∂
τ
)
+
1
sin
σ
(
cosh
τ
−
cos
σ
)
∂
2
Φ
∂
ϕ
2
]
{\displaystyle {\begin{aligned}\nabla ^{2}\Phi ={\frac {\left(\cosh \tau -\cos \sigma \right)^{3}}{a^{2}\sin \sigma }}&\left[{\frac {\partial }{\partial \sigma }}\left({\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \sigma }}\right)\right.\\[8pt]&{}\quad +\left.\sin \sigma {\frac {\partial }{\partial \tau }}\left({\frac {1}{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \tau }}\right)+{\frac {1}{\sin \sigma \left(\cosh \tau -\cos \sigma \right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}\right]\end{aligned}}}
Other differential operators such as
∇
⋅
F
{\displaystyle \nabla \cdot \mathbf {F} }
and
∇
×
F
{\displaystyle \nabla \times \mathbf {F} }
can be expressed in the coordinates
(
σ
,
τ
)
{\displaystyle (\sigma ,\tau )}
by substituting the scale factors into the general formulae found in orthogonal coordinates .
Applications
[ edit ]
The classic applications of bispherical coordinates are in solving partial differential equations ,
e.g., Laplace's equation , for which bispherical coordinates allow a
separation of variables . However, the Helmholtz equation is not separable in bispherical coordinates. A typical example would be the electric field surrounding two conducting spheres of different radii.
References
[ edit ]
Bibliography
[ edit ]
Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Parts I and II . New York: McGraw-Hill. pp. 665–666, 1298–1301.
Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers . New York: McGraw-Hill. p. 182. LCCN 59014456 .
Zwillinger D (1992). Handbook of Integration . Boston, MA: Jones and Bartlett. p. 113. ISBN 0-86720-293-9 .
Moon PH, Spencer DE (1988). "Bispherical Coordinates (η, θ, ψ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer Verlag. pp. 110–112 (Section IV, E4Rx). ISBN 0-387-02732-7 .
External links
[ edit ]
t
e
Two dimensional
Three dimensional
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Bispherical_coordinates&oldid=1196481289 "
C a t e g o r i e s :
● T h r e e - d i m e n s i o n a l c o o r d i n a t e s y s t e m s
● O r t h o g o n a l c o o r d i n a t e s y s t e m s
● T h i s p a g e w a s l a s t e d i t e d o n 1 7 J a n u a r y 2 0 2 4 , a t 1 7 : 0 3 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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