CircleBundleTransformationHyperbola2

1. Transforming lines to hyperbolas

Given two points {A(0,r),B(0,-r)}, taken on the y-axis symmetrically to the origin and a direction e(cos(u),sin(u)) define a transformation as follows:
- For each point X not lying on the x-axis consider the circle member c_X(x,y) of the circle bundle of all circles orthogonal to the circle (x²+y²)-r²=0 and the line x=0 (i.e. the bundle of non-intersecting type with limit-points {A(0,r),B(0,-r)}).
- Then construct Y to be the other intersection point of c_X with the line {X+te} through X and parallel to e.
The transformation Y=F(X) is well defined for every point of the plane except the x-axis. It is involutive (F² = 1) and has also the properties:

[1] F maps non-horizontal lines of the plane to hyperbolas.
[2] Let the line (v) be described through a vector equation X=a+tb, with (a=(a₁,0)) on the x-axis and (b) a unit vector. Then the hyperbola h=F(v) has asymptotes passing through {C(a₁,0),D(-a₁,0)} one of them being parallel to the direction of (e).
[3] The other asymptote of the hyperbola (h) passes through D(-a₁,0) and is inclined to the previous one by an angle equal to the angle of the line (v) to the x-axis. If E is the center of the hyperbola, the circumcircle of triangle CDE is tangent to (v).
[4] For each fixed direction (e) every circle of the bundle intersecting line (v) at points {X,X'} has corresponding images {Y=F(X),Y'=F(X')} so that chords YY' of the hyperbola are parallel. The direction of YY' and the horizontal line CD are equal inclined to an axis of the hyperbola.
[5] The hyperbolas resulting by fixing (v) and varying the direction (e) are tangent to two fixed circles {c₀,c₁} of the bundle at points {Y₀,Y₁}. The direction Y₀Y₁ is conjugate to the direction of YY'.
[6] Circles {c₀,c₁} and the circumcircle of CDE are all tangent to line (v) and by two define the same homothety center on (v). Line Y₀Y₁ passes through this homothety center.

The calculations of the file CircleBundleTransformation.html , handling the case of circle bundles of intersecting type, apply almost verbatim to this case of non-intersecting type. The only difference is the representation of the transformation Y=F(X), which now becomes:

This because the circle bundle can be described by the equation x²+y²+r² + ky =0, for variable k. This implies the previous formula and also the formulas (and formally results) of the aforementioned reference. In those formulas we have to replace only r² with -r².
The discussion then can continue as in CircleBundleTransformationHyperbola.html with the same change in the formulas and the same formal results about the asymptotes.

1. Transforming lines to hyperbolas

See Also