In mathematics, two-center bipolar coordinates is a coordinate system based on two coordinates which give distances from two fixed centers c 1 {\displaystyle c_{1}} and c 2 {\displaystyle c_{2}} .[1] This system is very useful in some scientific applications (e.g. calculating the electric field of a dipole on a plane).[2][3]
When the centers are at ( + a , 0 ) {\displaystyle (+a,0)} and ( − a , 0 ) {\displaystyle (-a,0)} , the transformation to Cartesian coordinates ( x , y ) {\displaystyle (x,y)} from two-center bipolar coordinates ( r 1 , r 2 ) {\displaystyle (r_{1},r_{2})} is
When x > 0, the transformation to polar coordinates from two-center bipolar coordinates is
where 2 a {\displaystyle 2a} is the distance between the poles (coordinate system centers).
Polar plotters use two-center bipolar coordinates to describe the drawing paths required to draw a target image.
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