双心坐标系

从直角坐标 ${\displaystyle (x,\ y)}$  变换到双心坐标 ${\displaystyle (r_{1},\ r_{2})}$ ^[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，双心坐标系的两个心点的坐标分别为${\displaystyle (-a,\ 0)}$  与 ${\displaystyle (+a,\ 0)}$  。

从极坐标 ${\displaystyle (r,\ \theta )}$  变换到双心坐标 ${\displaystyle (r_{1},\ r_{2})}$  ：

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

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

1. ^ ^{1.0 1.1} Weisstein, Eric W. (编). 雙心坐標系. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英语）. 
2. ^ R. Price, The Periodic Standing Wave Approximation: Adapted coordinates and spectral methods. (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.