# 雙極圓柱坐標系

## 基本定義

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

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

## 坐標曲面

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

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

### 逆變換

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

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

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

${\displaystyle \angle F_{1}PF_{2}}$ 是兩條從點 P 到兩個焦點的線段 ${\displaystyle {\overline {F_{1}P}}}$${\displaystyle {\overline {F_{2}P}}}$ 的夾角。這夾角的弧度是 ${\displaystyle \sigma }$ 。用餘弦定理來計算：

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

z-坐標的公式不變：

${\displaystyle z=z}$

## 標度因子

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

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

## 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.