CircleBundleTransformationParabola2

Transforming lines to parabolas II

For each point X not lying on the x-axis consider the circle member c_X(x,y) of the tangential bundle of all circles tangent to line y=0 at the origin O.
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 maps horizontal lines of the plane to parabolas. This was discussed in CircleBundleTransformationParabola.html for the more general case of the bundle of intersecting type with two base points on the x-axis and symmetric with respect to O. The present case is a limit case in which the two base points coincide. Here are some geometric properties of the parabolas thus defined.

[1] Let the line (v) be described through a vector equation X=a+tb, with (a=(0,a₂)) on the y-axis and (b) the unit vector (1,0). Then the parabola h=F(v) is tangent to the x-axis and has its axis parallel to the direction (e).
[2] The circle (c) of the bundle which is tangent to line v is also tangent to the parabola and their common tangent (g) is the reflexion of (v) with respect to the line (h) passing through the center of this circle and parallel to the orthogonal direction Je of e (J denotes the positive rotation by a right angle).
[3] The chords YY' intercepted by the circles of the bundle on the parabola are parallel and have conjugate diameter the line Aa (parallel to (e)).
[4] The focus of the parabola is easily determined by the circumcircle of the isosceles triangle BCD, C being the middle of OD and BC parallel to OA (or to Je).
Proofs are very similar to the ones contained in the aforementioned reference.
There is an analogous case for parabolas generated through circle-bundles of non-intersecting type studied in CircleBundleTransformationParabola3.html .

Transforming lines to parabolas II

See Also