# 圓環坐標系

## 數學定義

${\displaystyle x=a\ {\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}\cos \phi }$
${\displaystyle y=a\ {\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}\sin \phi }$
${\displaystyle z=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}}$

${\displaystyle \tau =\ln {\frac {d_{1}}{d_{2}}}}$

### 坐標曲面

${\displaystyle x^{2}+y^{2}+(z-a\cot \sigma )^{2}={\frac {a^{2}}{\sin ^{2}\sigma }}}$

${\displaystyle z^{2}+\left({\sqrt {x^{2}+y^{2}}}-a\coth \tau \right)^{2}={\frac {a^{2}}{\sinh ^{2}\tau }}}$

${\displaystyle \tau =0}$ 曲線與 z-軸同軸。當 ${\displaystyle \tau }$ 值增加時，圓球面的半徑會減少，圓球心會靠近焦點。

### 逆變換

${\displaystyle \tau }$${\displaystyle d_{1}}$${\displaystyle d_{2}}$ 的比例的自然對數

${\displaystyle \tau =\ln {\frac {d_{1}}{d_{2}}}}$

${\displaystyle \tan \phi ={\frac {y}{x}}}$

${\displaystyle d_{1}^{2}=({\sqrt {x^{2}+y^{2}}}+a)^{2}+z^{2}}$
${\displaystyle d_{2}^{2}=({\sqrt {x^{2}+y^{2}}}-a)^{2}+z^{2}}$

${\displaystyle \cos \sigma ={\frac {d_{1}^{2}+d_{2}^{2}-4a^{2}}{2d_{1}d_{2}}}}$

### 標度因子

${\displaystyle h_{\sigma }=h_{\tau }={\frac {a}{\cosh \tau -\cos \sigma }}}$

${\displaystyle h_{\phi }={\frac {a\sinh \tau }{\cosh \tau -\cos \sigma }}}$

${\displaystyle dV={\frac {a^{3}\sinh \tau }{\left(\cosh \tau -\cos \sigma \right)^{3}}}d\sigma d\tau d\phi }$
${\displaystyle \nabla ^{2}\Phi ={\frac {\left(\cosh \tau -\cos \sigma \right)^{3}}{a^{2}\sinh \tau }}\left[{\frac {\partial }{\partial \sigma }}\left({\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \sigma }}\right)+{\frac {\partial }{\partial \tau }}\left({\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \tau }}\right)+{\frac {1}{\sinh \tau \left(\cosh \tau -\cos \sigma \right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}\right]}$

## 參考文獻

• 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.

## 參考目錄

• 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.