it is relatively easy to show using inversion that, in the apollonian circles, every blue circle intersects every red circle orthogonally, i.e., at a right angle. inversion of the blue apollonian circles with respect to a circle centered on point "c" results in a pencil of concentric circles centered at the image of point "d". the same inversion transforms the red circles into a set of straight lines that all contain the image of "d". thus, this inversion transforms the bipolar coordinate system defined by the apollonian circles into a polar coordinate system.