# 雙極圓柱坐標系

## 基本定義

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

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

## 坐標曲面

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

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

### 逆變換

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

$\tau$$d_{1}$$d_{2}$ 的比例的自然對數

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

$\angle F_1PF_2$ 是兩條從點 P 到兩個焦點的線段 $\overline{F_1 P}$$\overline{F_2 P}$ 的夾角。這夾角的弧度是 $\sigma$ 。用餘弦定理來計算：

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

z-坐標的公式不變：

$z=z$

## 標度因子

$h_{\sigma} = h_{\tau} = \frac{a}{\cosh \tau - \cos\sigma}$
$h_{z}=1$

$dV = \frac{a^{2}}{\left( \cosh \tau - \cos\sigma \right)^{2}} d\sigma d\tau dz$
$\nabla^{2} \Phi = \frac{1}{a^{2}} \left( \cosh \tau - \cos\sigma \right)^{2} \left( \frac{\partial^{2} \Phi}{\partial \sigma^{2}} + \frac{\partial^{2} \Phi}{\partial \tau^{2}} \right) + \frac{\partial^{2} \Phi}{\partial z^{2}}$

## 參考文獻

• Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: pp. 187–190.
• Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 182. ASIN B0000CKZX7.
• Moon P, Spencer DE. Conical Coordinates (r, θ, λ). Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions corrected 2nd ed., 3rd print ed. New York: Springer-Verlag. 1988: unknown. ISBN 978-0387184302.