圖 1 ）圓環坐標系的幾個坐標曲面 。紅色圓球面的
σ
=
30
∘
{\displaystyle \sigma =30^{\circ }}
τ
=
0.5
{\displaystyle \tau =0.5}
ϕ
=
60
∘
{\displaystyle \phi =60^{\circ }}
直角坐標 大約為
(
0.996
,
−
1.725
,
1.911
)
{\displaystyle (0.996,\ -1.725,\ 1.911)}
圖 2 ）雙極坐標系繪圖。紅色圓圈變成上圖的紅色圓球面（
σ
{\displaystyle \sigma }
τ
{\displaystyle \tau }
圓環坐標系 （英語：Toroidal coordinates ）是一種三維正交坐標系 。設定二維橢圓坐標系 包含於 xz-平面；兩個焦點
F
1
{\displaystyle F_{1}}
F
2
{\displaystyle F_{2}}
直角坐標 分別為
(
−
a
,
0
,
0
)
{\displaystyle (-a,\ 0,\ 0)}
(
a
,
0
,
0
)
{\displaystyle (a,\ 0,\ 0)}
a
{\displaystyle a}
焦圓 ，又稱為參考圓 。
在三維空間裏，一個點 P 的圓環坐標
(
σ
,
τ
,
ϕ
)
{\displaystyle (\sigma ,\ \tau ,\ \phi )}
x
=
a
sinh
τ
cosh
τ
−
cos
σ
cos
ϕ
{\displaystyle x=a\ {\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}\cos \phi }
y
=
a
sinh
τ
cosh
τ
−
cos
σ
sin
ϕ
{\displaystyle y=a\ {\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}\sin \phi }
z
=
a
sin
σ
cosh
τ
−
cos
σ
{\displaystyle z=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}}
其中，
(
x
,
y
,
z
)
{\displaystyle (x,\ y,\ z)}
直角坐標 ，
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}}
τ
=
ln
d
1
d
2
{\displaystyle \tau =\ln {\frac {d_{1}}{d_{2}}}}
圓環坐標的值域為
−
π
<
σ
≤
π
{\displaystyle -\pi <\sigma \leq \pi }
τ
≥
0
{\displaystyle \tau \geq 0}
0
≤
ϕ
<
2
π
{\displaystyle 0\leq \phi <2\pi }
每一個
σ
{\displaystyle \sigma }
坐標曲面 都是包含了焦圓，而不同心的圓球面。圓球半徑為
x
2
+
y
2
+
(
z
−
a
cot
σ
)
2
=
a
2
sin
2
σ
{\displaystyle x^{2}+y^{2}+(z-a\cot \sigma )^{2}={\frac {a^{2}}{\sin ^{2}\sigma }}}
正值
σ
{\displaystyle \sigma }
σ
{\displaystyle \sigma }
|
σ
|
{\displaystyle \left|\sigma \right|}
|
σ
|
{\displaystyle \left|\sigma \right|}
π
/
2
{\displaystyle \pi /2}
每一個
τ
{\displaystyle \tau }
坐標曲面 都是不相交的環面。每一個環面都包圍著焦圓。環面半徑為
z
2
+
(
x
2
+
y
2
−
a
coth
τ
)
2
=
a
2
sinh
2
τ
{\displaystyle z^{2}+\left({\sqrt {x^{2}+y^{2}}}-a\coth \tau \right)^{2}={\frac {a^{2}}{\sinh ^{2}\tau }}}
τ
=
0
{\displaystyle \tau =0}
τ
{\displaystyle \tau }
圖 3 ）點 P 的坐標
σ
{\displaystyle \sigma }
τ
{\displaystyle \tau }
ϕ
{\displaystyle \phi }
F
1
P
¯
{\displaystyle {\overline {F_{1}P}}}
F
2
P
¯
{\displaystyle {\overline {F_{2}P}}}
∠
F
1
P
F
2
{\displaystyle \angle F_{1}PF_{2}}
σ
{\displaystyle \sigma }
F
1
P
{\displaystyle F_{1}P}
F
2
P
{\displaystyle F_{2}P}
自然對數 是
τ
{\displaystyle \tau }
σ
{\displaystyle \sigma }
τ
{\displaystyle \tau }
τ
{\displaystyle \tau }
d
1
{\displaystyle d_{1}}
d
2
{\displaystyle d_{2}}
自然對數 ：
τ
=
ln
d
1
d
2
{\displaystyle \tau =\ln {\frac {d_{1}}{d_{2}}}}
圓環坐標
(
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}}}
點 P 與兩個焦點之間的距離是
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}}
如圖 3 ，
∠
F
1
P
F
2
{\displaystyle \angle F_{1}PF_{2}}
F
1
P
¯
{\displaystyle {\overline {F_{1}P}}}
F
2
P
¯
{\displaystyle {\overline {F_{2}P}}}
σ
{\displaystyle \sigma }
餘弦定理 來計算：
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}}}}
圓環坐標
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 }}}
方位角的標度因子為
h
ϕ
=
a
sinh
τ
cosh
τ
−
cos
σ
{\displaystyle h_{\phi }={\frac {a\sinh \tau }{\cosh \tau -\cos \sigma }}}
無窮小體積元素是
d
V
=
a
3
sinh
τ
(
cosh
τ
−
cos
σ
)
3
d
σ
d
τ
d
ϕ
{\displaystyle dV={\frac {a^{3}\sinh \tau }{\left(\cosh \tau -\cos \sigma \right)^{3}}}d\sigma d\tau d\phi }
拉普拉斯算子 是
∇
2
Φ
=
(
cosh
τ
−
cos
σ
)
3
a
2
sinh
τ
[
∂
∂
σ
(
sinh
τ
cosh
τ
−
cos
σ
∂
Φ
∂
σ
)
+
∂
∂
τ
(
sinh
τ
cosh
τ
−
cos
σ
∂
Φ
∂
τ
)
+
1
sinh
τ
(
cosh
τ
−
cos
σ
)
∂
2
Φ
∂
ϕ
2
]
{\displaystyle \nabla ^{2}\Phi ={\frac {\left(\cosh \tau -\cos \sigma \right)^{3}}{a^{2}\sinh \tau }}\left[{\frac {\partial }{\partial \sigma }}\left({\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \sigma }}\right)+{\frac {\partial }{\partial \tau }}\left({\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \tau }}\right)+{\frac {1}{\sinh \tau \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)}
正交坐標系 條目內對應的一般公式。
圓環坐標有一個經典的應用，這是在解析像拉普拉斯方程 這類的偏微分方程式 。在這些方程式裏，圓環坐標允許分離變數法 的使用。個典型的例題是，有一個圓環導體 ，請問其周圍的電位 與電場 為什麼？應用圓環坐標，我們可以精緻地分析這例題。
由於托卡馬克 的圓環形狀，圓環坐標時常用在托卡馬克 的核融合 理論研究。
Arfken G. Mathematical Methods for Physicists 2nd ed. Orlando, FL: Academic Press. 1970: pp. 112–115. Andrews, Mark. Alternative separation of Laplace’s equation in toroidal coordinates and its application to electrostatics. Journal of Electrostatics. 2006, 64 : 664–672.
Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 666. Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 182. Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: pp. 190–192. 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. 112–115 (Section IV, E4Ry). ISBN 0-387-02732-7 
![{\displaystyle \nabla ^{2}\Phi ={\frac {\left(\cosh \tau -\cos \sigma \right)^{3}}{a^{2}\sinh \tau }}\left[{\frac {\partial }{\partial \sigma }}\left({\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \sigma }}\right)+{\frac {\partial }{\partial \tau }}\left({\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \tau }}\right)+{\frac {1}{\sinh \tau \left(\cosh \tau -\cos \sigma \right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b04cfc33fc0ce511686cc451a3c3f0b396b06e59)
