Apollonian circles

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Figure 1: Some Apollonian circles. Every blue circle intersects every red circle at a right angle, and vice versa of course. Every red circle passes through the two foci, C and D.

Apollonian circles are two families of circles such that every circle in the first family intersects every circle in the second family orthogonally. These circles form the basis for bipolar coordinates. They were discovered by Apollonius of Perga a renowned Greek geometer.

The Apollonian circles are defined by a line segment denoted CD. The circles of the first family (the blue circles of Figure 1) are defined by having different distance ratios to C and D; larger circles surround smaller circles but no circles are concentric. The circles of the second family (the red circles of Figure 1) all pass through the points C and D.

It is relatively easy to show that 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 set of concentric circles surrounding D', the image of point D. The same inversion transforms the red circles into a set of straight lines emanating from D'. Thus, this inversion transforms the bipolar coordinate system into a polar coordinate system. Since inversion is a conformal transformation, and every radial line intersects the concentric circles orthogonally, so do the original Apollonian circles.