雙心坐標系

直角座標系與極坐標系

從直角坐標 ${\displaystyle (x,\ y)}$ 變換到雙心坐標 ${\displaystyle (r_{1},\ r_{2})}$[1] ：

${\displaystyle x={\frac {r_{1}^{2}-r_{2}^{2}}{4a}}}$ 、

${\displaystyle y=\pm {\frac {1}{4a}}{\sqrt {16a^{2}r_{1}^{2}-(r_{1}^{2}-r_{2}^{2}+4a^{2})^{2}}}}$ ；

其中，雙心坐標系的兩個心點的坐標分別為${\displaystyle (-a,\ 0)}$ 與 ${\displaystyle (+a,\ 0)}$ 。

從極坐標 ${\displaystyle (r,\ \theta )}$ 變換到雙心坐標 ${\displaystyle (r_{1},\ r_{2})}$ ：

${\displaystyle r={\sqrt {\frac {r_{1}^{2}+r_{2}^{2}-2a^{2}}{2}}}}$ 、

${\displaystyle \theta =\arctan \left[{\frac {\sqrt {8a^{2}(r_{1}^{2}+r_{2}^{2}-2a^{2})-(r_{1}^{2}-r_{2}^{2})^{2}}}{r_{1}^{2}-r_{2}^{2}}}\right]}$ 。

1. Weisstein, Eric W. (编). . at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英语）.
2. (PDF). [2008-04-18]. （原始内容存档 (PDF)于2016-03-04）.
3. The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method.