Apollonian circles

They were discovered by Apollonius of Perga, a renowned ancient Greek geometer.

The Apollonian circles are defined in two different ways by a line segment denoted CD.

Each circle in the first family (the blue circles in the figure) is associated with a positive real number r, and is defined as the locus of points X such that the ratio of distances from X to C and to D equals r,

Each circle in the second family (the red circles in the figure) is associated with an angle θ, and is defined as the locus of points X such that the inscribed angle ∠CXD equals θ,

In order to obtain bipolar coordinates, a method is required to specify which point is the right one.

Such an arc is contained into a red circle and is bounded by points C, D. The remaining part of the corresponding red circle is isopt(θ + π).

When we really want the whole red circle, a description using oriented angles of straight lines has to be used:

Specifically, one is an elliptic pencil (red family of circles in the figure) that is defined by two generators that pass through each other in exactly two points (C, D).

The other is a hyperbolic pencil (blue family of circles in the figure) that is defined by two generators that do not intersect each other at any point.

[2] The elliptic pencil of circles passing through the two points C, D (the set of red circles, in the figure) has the line CD as its radical axis.

The centers of the circles in this pencil lie on the perpendicular bisector of CD.

The hyperbolic pencil defined by points C, D (the blue circles) has its radical axis on the perpendicular bisector of line CD, and all its circle centers on line CD.

Alternatively,[3] the orthogonal property of the two pencils follows from the defining property of the radical axis, that from any point X on the radical axis of a pencil P the lengths of the tangents from X to each circle in P are all equal.

The pencil of circles perpendicular to a parabolic pencil is also parabolic; it consists of the circles that have the same common tangent point but with a perpendicular tangent line at that point.

[4] Apollonian trajectories have been shown to be followed in their motion by vortex cores or other defined pseudospin states in some physical systems involving interferential or coupled fields, such photonic or coupled polariton waves.

[5] The trajectories arise from the Rabi rotation of the Bloch sphere and its stereographic projection on the real space where the observation is made.