# 圓環坐標系

## 數學定義

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

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

### 坐標曲面

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

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

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

### 逆變換

$\tau$$d_{1}$$d_{2}$ 的比例的自然對數

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

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

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

$\cos \sigma =\frac{d_1^2 + d_2^2 - 4a^2}{2 d_1 d_2}$

### 標度因子

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

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

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

## 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.