在三维空间里，一个点 P 的双球坐标
(
σ
,
τ
,
ϕ
)
{\displaystyle (\sigma ,\ \tau ,\ \phi )}
x
=
a
sin
σ
cosh
τ
−
cos
σ
cos
ϕ
{\displaystyle x=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}\cos \phi }
y
=
a
sin
σ
cosh
τ
−
cos
σ
sin
ϕ
{\displaystyle y=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}\sin \phi }
z
=
a
sinh
τ
cosh
τ
−
cos
σ
{\displaystyle z=a\ {\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}}
其中，
(
x
,
y
,
z
)
{\displaystyle (x,\ y,\ z)}
直角坐标 ，
σ
{\displaystyle \sigma }
∠
F
1
P
F
2
{\displaystyle \angle F_{1}PF_{2}}
弧度 ，
τ
{\displaystyle \tau }
d
1
{\displaystyle d_{1}}
d
2
{\displaystyle d_{2}}
自然对数 ：
τ
=
ln
d
1
d
2
{\displaystyle \tau =\ln {\frac {d_{1}}{d_{2}}}}
每一个红色的
σ
{\displaystyle \sigma }
坐标曲面 都是包含了两个焦点
F
1
{\displaystyle F_{1}}
F
2
{\displaystyle F_{2}}
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 }}}
当绝对值
|
σ
|
{\displaystyle \left|\sigma \right|}
|
σ
|
{\displaystyle \left|\sigma \right|}
π
/
2
{\displaystyle \pi /2}
每一个蓝色的
τ
{\displaystyle \tau }
坐标曲面 都是不相交的圆球面。每一个圆球面都包围著一个焦点；圆球心都包含于 z-轴。圆球半径为
x
2
+
y
2
+
(
z
−
a
coth
τ
)
2
=
a
2
sinh
2
τ
{\displaystyle x^{2}+y^{2}+(z-a\coth \tau )^{2}={\frac {a^{2}}{\sinh ^{2}\tau }}}
它们的圆球心都包含于 z-轴。正值
τ
{\displaystyle \tau }
z
>
0
{\displaystyle z>0}
τ
{\displaystyle \tau }
z
<
0
{\displaystyle z<0}
τ
=
0
{\displaystyle \tau =0}
τ
{\displaystyle \tau }
图 3 ）点 P 的坐标
σ
{\displaystyle \sigma }
τ
{\displaystyle \tau }
ϕ
{\displaystyle \phi }
σ
{\displaystyle \sigma }
∠
F
1
P
F
2
{\displaystyle \angle F_{1}PF_{2}}
τ
{\displaystyle \tau }
d
1
{\displaystyle d_{1}}
d
2
{\displaystyle d_{2}}
自然对数 。
σ
{\displaystyle \sigma }
τ
{\displaystyle \tau }
双球坐标
(
σ
,
τ
,
ϕ
)
{\displaystyle (\sigma ,\ \tau ,\ \phi )}
(
x
,
y
,
z
)
{\displaystyle (x,\ y,\ z)}
ϕ
{\displaystyle \phi }
tan
ϕ
=
y
x
{\displaystyle \tan \phi ={\frac {y}{x}}}
点 P 与两个焦点之间的距离是
d
1
2
=
x
2
+
y
2
+
(
z
+
a
)
2
{\displaystyle d_{1}^{2}=x^{2}+y^{2}+(z+a)^{2}}
d
2
2
=
x
2
+
y
2
+
(
z
−
a
)
2
{\displaystyle d_{2}^{2}=x^{2}+y^{2}+(z-a)^{2}}
τ
{\displaystyle \tau }
d
1
{\displaystyle d_{1}}
d
2
{\displaystyle d_{2}}
自然对数 ：
τ
=
ln
d
1
d
2
{\displaystyle \tau =\ln {\frac {d_{1}}{d_{2}}}}
如图 3 ，
∠
F
1
P
F
2
{\displaystyle \angle F_{1}PF_{2}}
σ
{\displaystyle \sigma }
馀弦定理 来计算：
cos
σ
=
d
1
2
+
d
2
2
−
4
a
2
2
d
1
d
2
{\displaystyle \cos \sigma ={\frac {d_{1}^{2}+d_{2}^{2}-4a^{2}}{2d_{1}d_{2}}}}
双球坐标
σ
{\displaystyle \sigma }
τ
{\displaystyle \tau }
h
σ
=
h
τ
=
a
cosh
τ
−
cos
σ
{\displaystyle h_{\sigma }=h_{\tau }={\frac {a}{\cosh \tau -\cos \sigma }}}
方位角的标度因子为
h
ϕ
=
a
sin
σ
cosh
τ
−
cos
σ
{\displaystyle h_{\phi }={\frac {a\sin \sigma }{\cosh \tau -\cos \sigma }}}
无穷小体积元素是
d
V
=
a
3
sin
σ
(
cosh
τ
−
cos
σ
)
3
d
σ
d
τ
d
ϕ
{\displaystyle dV={\frac {a^{3}\sin \sigma }{(\cosh \tau -\cos \sigma )^{3}}}d\sigma d\tau d\phi }
拉普拉斯算子 是
∇
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 \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)+\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]}
其它微分算子，像
∇
⋅
F
{\displaystyle \nabla \cdot \mathbf {F} }
∇
×
F
{\displaystyle \nabla \times \mathbf {F} }
(
σ
,
τ
,
z
)
{\displaystyle (\sigma ,\ \tau ,\ z)}
正交坐标系 条目内对应的一般公式。
Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 665–666. Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 182. Zwillinger D. Handbook of Integration. Boston, MA: Jones and Bartlett. 1992: p. 113. ISBN 0-86720-293-9 Moon PH, Spencer DE. Toroidal Coordinates (η, θ, ψ). Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions 2nd ed., 3rd revised printing. New York: Springer Verlag. 1988: pp. 110–112 (Section IV, E4Ry). ISBN 0-387-02732-7 
![{\displaystyle \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)+\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4cf647df653a77be2f7d81a9ffc2c3c0d31820c)
