title: Two-center bipolar coordinates
text: # Two-center bipolar coordinates
  
  In mathematics, two-center bipolar coordinates is a coordinate system based on two coordinates which give distances from two fixed centers $c_{1}$ and $c_{2}$. This system is very useful in some scientific applications (e.g. calculating the electric field of a dipole on a plane).
  
  ## Transformation to Cartesian coordinates
  
  When the centers are at $(+a,0)$ and $(-a,0)$, the transformation to Cartesian coordinates $(x,y)$ from two-center bipolar coordinates $(r_{1},r_{2})$ is
  
      $x={\frac {r_{2}^{2}-r_{1}^{2}}{4a}}$
      $y=\pm {\frac {1}{4a}}{\sqrt {16a^{2}r_{2}^{2}-(r_{2}^{2}-r_{1}^{2}+4a^{2})^{2}}}$
  
  ## Transformation to polar coordinates
  
  When x > 0, the transformation to polar coordinates from two-center bipolar coordinates is
  
      $r={\sqrt {\frac {r_{1}^{2}+r_{2}^{2}-2a^{2}}{2}}}$
      $\theta =\arctan \left({\frac {\sqrt {r_{1}^{4}-8a^{2}r_{1}^{2}-2r_{1}^{2}r_{2}^{2}-(4a^{2}-r_{2}^{2})^{2}}}{r_{2}^{2}-r_{1}^{2}}}\right)$
  
  where $2a$ is the distance between the poles (coordinate system centers).
  
  ## Applications
  
  Polar plotters use two-center bipolar coordinates to describe the drawing paths required to draw a target image.
  
  