ApollonianBundle

1. Apollonian Bundle

Consider a segment a = EG and the Apollonian circles dividing it in various ratios (see the file Apollonian_Circles.html ). All these circles are members of a hyperbolic circle bundle which I call the Apollonian Bundle of the segment.
The common radical axis of the circles-members of the bundle is the medial line of the segment.
As with every bundle, the Apollonian bundle is generated from two particular members, one of which can be the medial line (e) of the segment.
Consider a circle-member (f) of the bundle dividing in the ratio k (blue). Then the member of the bundle, passing trhough the center K of (f) (the gold circle c) divides in the ratio k². In the figure below the ratio k is defined by the lengths of two segments k = |AB|/|CD|.

2. Properties

1) Every hyperbolic circle-bundle is the Apollonian Bundle of the segment defined by its limiting points     (corresponding to E, G above).
2) The circles of the bundle can be parameterized by their ratio k=|FE|/|FG|.
3) The members left from line (e) are those for which k<1.
4) The members right from (e) are those for which k>1.
5) Two members with inverse ratios k*k' = 1 are symmetric w.r. to (e).
6) A half-circle with diameter EG gives another parametrization of the bundle, in the following sense.
    6.1) Every member of the bundle cuts it in exactly one point P and k = |PE|/|PG| = tan(phi).
    6.2) This gives another way to construct the circle corresponding to ratio k:
    6.2) Find (phi) such that  phi = arctan(k) and locate point P on the half circle on EG.
    6.3) Draw PK orthogonal to OP at P. This determines the center K and the radius |KP| of the k-ratio-circle.
7) Notice that the aforementioned half circle is part of the minimal circle-member of the bundle which     is orthogonal to the Apollonian one. Hence the tangency of OP at circle (f).
8) Points {W,W*} are harmonic conjugate with respect to {E,G}. Their cross-ratio (E,G,W,W*)=(WE/WG):(W*E/W*G)=-1. Inversely every pair of harmonic conjugate points with respect to {E,G} defines a circle of the Apollonian bundle and a cross ratio (E,G,W,W*)=-1.

3. Formularium

Given a segment OA of length (coordinate) a, and a ratio (k<0 say),
1) the coordinate of X such that XO/XA = k is at  x = ka/(k-1),
2) the coordinate of X' such that XO/XA = -k is at x' = ka/(k+1),
3) the center of the Apollonian circle is at   (x+x')/2 = ak²/(k²-1),
4) the distance (or diameter of Apollonian circle)  is     XX'  = -2ka/(k²-1).

1. Apollonian Bundle

2. Properties

3. Formularium

See Also