# 雙極坐標系

## 基本定義

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

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

（回想 ${\displaystyle F_{1}\,\!}$ ${\displaystyle F_{2}\,\!}$  的坐標分別為 ${\displaystyle (-a,\ 0)\,\!}$ ${\displaystyle (a,\ 0)\,\!}$  ）。

## 等值曲線

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

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

### 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：

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

## 標度因子

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

${\displaystyle dA={\frac {a^{2}}{(\cosh \tau -\cos \sigma )^{2}}}\ d\sigma d\tau \,\!}$
${\displaystyle \nabla ^{2}\Phi =\left({\frac {\cosh \tau -\cos \sigma }{a}}\right)^{2}({\frac {\partial ^{2}\Phi }{\partial \sigma ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \tau ^{2}}})\,\!}$

## 參考文獻

• H. Bateman "Spheroidal and bipolar coordinates", Duke Mathematical Journal 4 (1938), no. 1, 39–50。
• Lockwood, E. H. "Bipolar Coordinates." Chapter 25 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 186-190, 1967。
• Korn GA and Korn TM, (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill。