# 雙極坐標系

## 基本定義

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

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

（回想 $F_1\,\!$$F_2\,\!$ 的坐標分別為 $( - a,\ 0)\,\!$$(a,\ 0)\,\!$ ）。

## 等值曲線

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

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

### 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：

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

## 標度因子

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

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

## 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。