图1）扁球面坐标系的几个坐标曲面 。红色扁球面的
μ
=
1
{\displaystyle \mu =1}
ν
=
45
∘
{\displaystyle \nu =45^{\circ }}
ϕ
=
−
60
∘
{\displaystyle \phi =-60^{\circ }}
二面角 角度是
−
60
∘
{\displaystyle -60^{\circ }}
直角坐标 大约为
(
1.09
,
−
1.89
,
1.66
)
{\displaystyle (1.09,\ -1.89,\ 1.66)}
图2）椭圆坐标系绘图。横轴是x-轴，竖轴是z-轴。红色椭圆（
μ
{\displaystyle \mu }
μ
{\displaystyle \mu }
x
>
0
{\displaystyle x>0}
ν
{\displaystyle \nu }
ν
{\displaystyle \nu }
扁球面坐标系 （英语：Oblate spheroidal coordinates ）是一种三维正交坐标系 。设定二维椭圆坐标系 包含于xz-平面；两个焦点
F
1
{\displaystyle F_{1}}
F
2
{\displaystyle F_{2}}
直角坐标 分别为
(
−
a
,
0
,
0
)
{\displaystyle (-a,\ 0,\ 0)}
(
a
,
0
,
0
)
{\displaystyle (a,\ 0,\ 0)}
长球面坐标系 。）椭圆坐标系的两个焦点，变为一个半径为
a
{\displaystyle a}
焦圆 ，又称为参考圆 。扁球面坐标系可以被视为椭球坐标系 的极限案例，其两个最大的半轴的长度相同。
当边界条件涉及扁球面 或旋转双曲面 时，扁球面坐标时常可以用来解析偏微分方程式 。例如，关于佩兰摩擦因子 （Perrin friction factors ）的计算，扁球面坐标扮演了极重要的角色。让·佩兰 因此而荣获1926年诺贝尔物理奖 。佩兰摩擦因子决定了分子 的旋转扩散 （rotational diffusion ）。这程序又影响了许多科技，像蛋白质 核磁共振 光谱学 （protein NMR ），的可行性。应用这程序，我们可以推论分子的流体动力 体积与形状。扁球面坐标也时常用来解析电磁学 （例如，扁球形带电的分子的电容率 ），声学 （例如，声音通过圆孔时产生的散射），流体动力学 （水通过消防水带的喷口），扩散理论 （红热的钱币在水里的冷却），等等方面的问题。
在三维空间里，一个点P的扁球面坐标
(
μ
,
ν
,
ϕ
)
{\displaystyle (\mu ,\ \nu ,\ \phi )}
x
=
a
cosh
μ
cos
ν
cos
ϕ
{\displaystyle x=a\ \cosh \mu \ \cos \nu \ \cos \phi }
y
=
a
cosh
μ
cos
ν
sin
ϕ
{\displaystyle y=a\ \cosh \mu \ \cos \nu \ \sin \phi }
z
=
a
sinh
μ
sin
ν
{\displaystyle z=a\ \sinh \mu \ \sin \nu }
其中，
μ
≥
0
{\displaystyle \mu \geq 0}
−
90
∘
≤
ν
≤
90
∘
{\displaystyle -90^{\circ }\leq \nu \leq 90^{\circ }}
−
180
∘
≤
ϕ
≤
180
∘
{\displaystyle -180^{\circ }\leq \phi \leq 180^{\circ }}
学术界比较中意这一种扁球面坐标，因为没有简并 ；三维空间内每一点都拥有自己独特的扁球面坐标。
μ
{\displaystyle \mu }
x
2
+
y
2
a
2
cosh
2
μ
+
z
2
a
2
sinh
2
μ
=
cos
2
ν
+
sin
2
ν
=
1
{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}\cosh ^{2}\mu }}+{\frac {z^{2}}{a^{2}\sinh ^{2}\mu }}=\cos ^{2}\nu +\sin ^{2}\nu =1}
它们是由椭圆绕著z-轴旋转形成的。椭球面与xz-平面的相交，是一个的椭圆。沿著x-轴，长半轴长度为
a
cosh
μ
{\displaystyle a\cosh \mu }
a
sinh
μ
{\displaystyle a\sinh \mu }
±
a
{\displaystyle \pm a}
ν
{\displaystyle \nu }
x
2
+
y
2
a
2
cos
2
ν
−
z
2
a
2
sin
2
ν
=
cosh
2
μ
−
sinh
2
μ
=
1
{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}\cos ^{2}\nu }}-{\frac {z^{2}}{a^{2}\sin ^{2}\nu }}=\cosh ^{2}\mu -\sinh ^{2}\mu =1}
假若
ν
{\displaystyle \nu }
z
{\displaystyle z}
ν
{\displaystyle \nu }
渐近线 的角度。所有双曲线的焦点都在x-轴，x-坐标分别为
±
a
{\displaystyle \pm a}
ϕ
{\displaystyle \phi }
x
sin
ϕ
−
y
cos
ϕ
=
0
{\displaystyle x\sin \phi -y\cos \phi =0}
用直角坐标
(
x
,
y
,
z
)
{\displaystyle (x,\ y,\ z)}
(
μ
,
ν
,
ϕ
)
{\displaystyle (\mu ,\ \nu ,\ \phi )}
ϕ
{\displaystyle \phi }
tan
ϕ
=
y
x
{\displaystyle \tan \phi ={\frac {y}{x}}}
设定
d
1
{\displaystyle d_{1}}
d
2
{\displaystyle d_{2}}
d
1
2
=
(
x
2
+
y
2
+
a
)
2
+
z
2
{\displaystyle d_{1}^{2}=({\sqrt {x^{2}+y^{2}}}+a)^{2}+z^{2}}
d
2
2
=
(
x
2
+
y
2
−
a
)
2
+
z
2
{\displaystyle d_{2}^{2}=({\sqrt {x^{2}+y^{2}}}-a)^{2}+z^{2}}
坐标
μ
{\displaystyle \mu }
ν
{\displaystyle \nu }
cosh
μ
=
d
1
+
d
2
2
a
{\displaystyle \cosh \mu ={\frac {d_{1}+d_{2}}{2a}}}
cos
ν
=
d
1
−
d
2
2
a
{\displaystyle \cos \nu ={\frac {d_{1}-d_{2}}{2a}}}
扁球面坐标
μ
{\displaystyle \mu }
ν
{\displaystyle \nu }
h
μ
=
h
ν
=
a
sinh
2
μ
+
sin
2
ν
{\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {\sinh ^{2}\mu +\sin ^{2}\nu }}}
方位角
ϕ
{\displaystyle \phi }
h
ϕ
=
a
cosh
μ
cos
ν
{\displaystyle h_{\phi }=a\cosh \mu \ \cos \nu }
无穷小体积元素是
d
V
=
a
3
cosh
μ
cos
ν
(
sinh
2
μ
+
sin
2
ν
)
d
μ
d
ν
d
ϕ
{\displaystyle dV=a^{3}\cosh \mu \ \cos \nu \ \left(\sinh ^{2}\mu +\sin ^{2}\nu \right)d\mu d\nu d\phi }
拉普拉斯算子 是
∇
2
Φ
=
1
a
2
(
sinh
2
μ
+
sin
2
ν
)
[
1
cosh
μ
∂
∂
μ
(
cosh
μ
∂
Φ
∂
μ
)
+
1
cos
ν
∂
∂
ν
(
cos
ν
∂
Φ
∂
ν
)
]
+
1
a
2
(
cosh
2
μ
cos
2
ν
)
∂
2
Φ
∂
ϕ
2
{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}\left[{\frac {1}{\cosh \mu }}{\frac {\partial }{\partial \mu }}\left(\cosh \mu {\frac {\partial \Phi }{\partial \mu }}\right)+{\frac {1}{\cos \nu }}{\frac {\partial }{\partial \nu }}\left(\cos \nu {\frac {\partial \Phi }{\partial \nu }}\right)\right]+{\frac {1}{a^{2}\left(\cosh ^{2}\mu \cos ^{2}\nu \right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}
其它微分算子，像
∇
⋅
F
{\displaystyle \nabla \cdot \mathbf {F} }
∇
×
F
{\displaystyle \nabla \times \mathbf {F} }
(
μ
,
ν
,
ϕ
)
{\displaystyle (\mu ,\ \nu ,\ \phi )}
正交坐标系 条目内对应的一般公式。
另外有一组有时会用到的扁球面坐标
(
ζ
,
ξ
,
ϕ
)
{\displaystyle (\zeta ,\ \xi ,\ \phi )}
ζ
=
sinh
μ
{\displaystyle \zeta =\sinh \mu }
ξ
=
sin
ν
{\displaystyle \xi =\sin \nu }
[ 1]
ζ
{\displaystyle \zeta }
ξ
{\displaystyle \xi }
x
=
a
(
1
+
ζ
2
)
(
1
−
ξ
2
)
cos
ϕ
{\displaystyle x=a{\sqrt {(1+\zeta ^{2})(1-\xi ^{2})}}\,\cos \phi }
y
=
a
(
1
+
ζ
2
)
(
1
−
ξ
2
)
sin
ϕ
{\displaystyle y=a{\sqrt {(1+\zeta ^{2})(1-\xi ^{2})}}\,\sin \phi }
z
=
a
ζ
ξ
{\displaystyle z=a\zeta \xi }
其中，实数
0
≤
ζ
<
∞
{\displaystyle 0\leq \zeta <\infty }
−
1
≤
ξ
<
1
{\displaystyle -1\leq \xi <1}
−
180
∘
≤
ϕ
≤
180
∘
{\displaystyle -180^{\circ }\leq \phi \leq 180^{\circ }}
扁球面坐标
(
ζ
,
ξ
,
ϕ
)
{\displaystyle (\zeta ,\ \xi ,\ \phi )}
h
ζ
=
a
ζ
2
+
ξ
2
1
+
ζ
2
{\displaystyle h_{\zeta }=a{\sqrt {\frac {\zeta ^{2}+\xi ^{2}}{1+\zeta ^{2}}}}}
h
ξ
=
a
ζ
2
+
ξ
2
1
−
ξ
2
{\displaystyle h_{\xi }=a{\sqrt {\frac {\zeta ^{2}+\xi ^{2}}{1-\xi ^{2}}}}}
h
ϕ
=
a
(
1
+
ζ
2
)
(
1
−
ξ
2
)
{\displaystyle h_{\phi }=a{\sqrt {(1+\zeta ^{2})(1-\xi ^{2})}}}
无穷小体积元素是
d
V
=
a
3
(
ζ
2
+
ξ
2
)
d
ζ
d
ξ
d
ϕ
{\displaystyle dV=a^{3}(\zeta ^{2}+\xi ^{2})\,d\zeta \,d\xi \,d\phi }
拉普拉斯算子 是
∇
2
V
=
1
a
2
(
ζ
2
+
ξ
2
)
{
∂
∂
ζ
[
(
1
+
ζ
2
)
∂
V
∂
ζ
]
+
∂
∂
ξ
[
(
1
−
ξ
2
)
∂
V
∂
ξ
]
}
+
1
a
2
(
1
+
ζ
2
)
(
1
−
ξ
2
)
∂
2
V
∂
ϕ
2
{\displaystyle \nabla ^{2}V={\frac {1}{a^{2}\left(\zeta ^{2}+\xi ^{2}\right)}}\left\{{\frac {\partial }{\partial \zeta }}\left[\left(1+\zeta ^{2}\right){\frac {\partial V}{\partial \zeta }}\right]+{\frac {\partial }{\partial \xi }}\left[\left(1-\xi ^{2}\right){\frac {\partial V}{\partial \xi }}\right]\right\}+{\frac {1}{a^{2}\left(1+\zeta ^{2}\right)\left(1-\xi ^{2}\right)}}{\frac {\partial ^{2}V}{\partial \phi ^{2}}}}
图3）第三种扁球面坐标系
(
σ
,
τ
,
ϕ
)
{\displaystyle (\sigma ,\ \tau ,\ \phi )}
坐标曲面 。红色扁球面是
σ
{\displaystyle \sigma }
τ
{\displaystyle \tau }
ϕ
{\displaystyle \phi }
二面角 角度是
ϕ
{\displaystyle \phi }
1 ，P2 （以黑色的圆球表示）观察到。 另外，还有一种比较有几何直觉性的扁球面坐标系
(
σ
,
τ
,
ϕ
)
{\displaystyle (\sigma ,\ \tau ,\ \phi )}
[ 2]
σ
=
cosh
μ
{\displaystyle \sigma =\cosh \mu }
τ
=
cos
ν
{\displaystyle \tau =\cos \nu }
ϕ
=
ϕ
{\displaystyle \phi =\phi }
坐标
σ
{\displaystyle \sigma }
τ
{\displaystyle \tau }
σ
{\displaystyle \sigma }
τ
{\displaystyle \tau }
ν
{\displaystyle \nu }
(
x
,
y
,
±
z
)
{\displaystyle (x,\ y,\ \pm z)}
(
σ
,
τ
,
ϕ
)
{\displaystyle (\sigma ,\ \tau ,\ \phi )}
x
=
a
σ
τ
cos
ϕ
{\displaystyle x=a\sigma \tau \cos \phi }
y
=
a
σ
τ
sin
ϕ
{\displaystyle y=a\sigma \tau \sin \phi }
z
2
=
a
2
(
σ
2
−
1
)
(
1
−
τ
2
)
{\displaystyle z^{2}=a^{2}\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}
坐标
σ
{\displaystyle \sigma }
τ
{\displaystyle \tau }
d
1
{\displaystyle d_{1}}
d
2
{\displaystyle d_{2}}
d
1
+
d
2
=
2
a
σ
{\displaystyle d_{1}+d_{2}=2a\sigma }
d
1
−
d
2
=
2
a
τ
{\displaystyle d_{1}-d_{2}=2a\tau }
所以，点P与焦圆的最远距离是
a
(
σ
+
τ
)
{\displaystyle a(\sigma +\tau )}
a
(
σ
−
τ
)
{\displaystyle a(\sigma -\tau )}
σ
{\displaystyle \sigma }
x
2
+
y
2
a
2
σ
2
+
z
2
a
2
(
σ
2
−
1
)
=
1
{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}\sigma ^{2}}}+{\frac {z^{2}}{a^{2}\left(\sigma ^{2}-1\right)}}=1}
τ
{\displaystyle \tau }
x
2
+
y
2
a
2
τ
2
−
z
2
a
2
(
1
−
τ
2
)
=
1
{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}\tau ^{2}}}-{\frac {z^{2}}{a^{2}\left(1-\tau ^{2}\right)}}=1}
ϕ
{\displaystyle \phi }
x
sin
ϕ
−
y
cos
ϕ
=
0
{\displaystyle x\sin \phi -y\cos \phi =0}
扁球面坐标
(
σ
,
τ
,
ϕ
)
{\displaystyle (\sigma ,\ \tau ,\ \phi )}
h
σ
=
a
σ
2
+
τ
2
σ
2
+
1
{\displaystyle h_{\sigma }=a{\sqrt {\frac {\sigma ^{2}+\tau ^{2}}{\sigma ^{2}+1}}}}
h
τ
=
a
σ
2
+
τ
2
1
−
τ
2
{\displaystyle h_{\tau }=a{\sqrt {\frac {\sigma ^{2}+\tau ^{2}}{1-\tau ^{2}}}}}
h
ϕ
=
a
σ
τ
{\displaystyle h_{\phi }=a\sigma \tau }
无穷小体积元素是
d
V
=
a
3
σ
τ
σ
2
+
τ
2
(
σ
2
+
1
)
(
1
−
τ
2
)
d
σ
d
τ
d
ϕ
{\displaystyle dV=a^{3}\sigma \tau {\frac {\sigma ^{2}+\tau ^{2}}{\sqrt {\left(\sigma ^{2}+1\right)\left(1-\tau ^{2}\right)}}}d\sigma d\tau d\phi }
拉普拉斯算子 是
∇
2
Φ
=
1
a
2
(
σ
2
+
τ
2
)
{
∂
∂
σ
[
(
σ
2
+
1
)
∂
Φ
∂
σ
]
+
∂
∂
τ
[
(
1
−
τ
2
)
∂
Φ
∂
τ
]
}
+
1
a
2
(
σ
2
+
1
)
(
1
−
τ
2
)
∂
2
Φ
∂
ϕ
2
{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sigma ^{2}+\tau ^{2}\right)}}\left\{{\frac {\partial }{\partial \sigma }}\left[\left(\sigma ^{2}+1\right){\frac {\partial \Phi }{\partial \sigma }}\right]+{\frac {\partial }{\partial \tau }}\left[\left(1-\tau ^{2}\right){\frac {\partial \Phi }{\partial \tau }}\right]\right\}+{\frac {1}{a^{2}\left(\sigma ^{2}+1\right)\left(1-\tau ^{2}\right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}
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。
如同球坐标 解答的形式为球谐函数 ，拉普拉斯方程 可以用分离变数法来求解，得到形式为扁球谐函数 的答案。假若，边界条件涉及扁球面，我们可以优先选择这方法来解析。
^ Smythe, 1968。
^ Abramowitz and Stegun, p. 752。
Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 662.
ξ
1
=
a
sinh
μ
{\displaystyle \xi _{1}=a\sinh \mu }
ξ
2
=
sin
ν
{\displaystyle \xi _{2}=\sin \nu }
ξ
3
=
cos
ϕ
{\displaystyle \xi _{3}=\cos \phi }
Zwillinger D. Handbook of Integration. Boston, MA: Jones and Bartlett. 1992: p. 115. ISBN 0-86720-293-9
u
k
{\displaystyle u_{k}}
ξ
k
{\displaystyle \xi _{k}}
Smythe, WR. Static and Dynamic Electricity 3rd ed. New York: McGraw-Hill. 1968. Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: p. 98.
ξ
=
a
sinh
μ
{\displaystyle \xi =a\sinh \mu }
η
=
sin
ν
{\displaystyle \eta =\sin \nu }
ϕ
=
ϕ
{\displaystyle \phi =\phi }
Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 177.
(
μ
,
ν
,
ϕ
)
{\displaystyle (\mu ,\ \nu ,\ \phi )}
(
σ
,
τ
,
ϕ
)
{\displaystyle (\sigma ,\ \tau ,\ \phi )}
Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: p. 182. 馀纬度
θ
=
90
∘
−
ν
{\displaystyle \theta =90^{\circ }-\nu }
纬度
ν
{\displaystyle \nu }
Moon PH, Spencer DE. Oblate spheroidal 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. 1988: pp. 31–34 (Table 1.07). ISBN 0-387-02732-7
θ
=
90
∘
−
ν
{\displaystyle \theta =90^{\circ }-\nu }
ϕ
{\displaystyle \phi }
ψ
{\displaystyle \psi }
Landau LD, Lifshitz EM, Pitaevskii LP. Electrodynamics of Continuous Media (Volume 8 of the Course of Theoretical Physics) 2nd edition. New York: Pergamon Press. 1984: pp. 19–29. ISBN 978-0750626347 
![{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}\left[{\frac {1}{\cosh \mu }}{\frac {\partial }{\partial \mu }}\left(\cosh \mu {\frac {\partial \Phi }{\partial \mu }}\right)+{\frac {1}{\cos \nu }}{\frac {\partial }{\partial \nu }}\left(\cos \nu {\frac {\partial \Phi }{\partial \nu }}\right)\right]+{\frac {1}{a^{2}\left(\cosh ^{2}\mu \cos ^{2}\nu \right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c49d17ab5b4a4dcede10f4ac4baf579dd5a32f86)
![{\displaystyle \nabla ^{2}V={\frac {1}{a^{2}\left(\zeta ^{2}+\xi ^{2}\right)}}\left\{{\frac {\partial }{\partial \zeta }}\left[\left(1+\zeta ^{2}\right){\frac {\partial V}{\partial \zeta }}\right]+{\frac {\partial }{\partial \xi }}\left[\left(1-\xi ^{2}\right){\frac {\partial V}{\partial \xi }}\right]\right\}+{\frac {1}{a^{2}\left(1+\zeta ^{2}\right)\left(1-\xi ^{2}\right)}}{\frac {\partial ^{2}V}{\partial \phi ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b357b2c9bd2183b7e40d3f74589b6b2b948608de)
![{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sigma ^{2}+\tau ^{2}\right)}}\left\{{\frac {\partial }{\partial \sigma }}\left[\left(\sigma ^{2}+1\right){\frac {\partial \Phi }{\partial \sigma }}\right]+{\frac {\partial }{\partial \tau }}\left[\left(1-\tau ^{2}\right){\frac {\partial \Phi }{\partial \tau }}\right]\right\}+{\frac {1}{a^{2}\left(\sigma ^{2}+1\right)\left(1-\tau ^{2}\right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7322471d8b4075c27b723835edf72369a8a1f85b)
