CircleBundleTransformationParabola

Transforming lines to parabolas

Given two points {A(r,0),B(-r,0)}, taken on the x-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 bundle (of all circles through the two points) generated by the circle (x²+y²)-r²=0 and the line y=0.
- 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 proved in CircleBundleTransformation.html .
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) passes through {A,B} and has its axis parallel to the direction (e).
[2] Chords YY' intercepted by the circles of the bundle on the parabola are parallel and their conjugate direction (containing their middles) is line CD (parallel to e).
[3] The circle (c) of the bundle passing through points {A,B,C} 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).

The first property, about the axis, was proved in the aforementioned reference. The second is a consequence of the parallelity of chords YY' intercepted by the circles of the bundle (see [5] of PowerGeneral.html ) is easily seen by considering the intersection points {X,X'} of line (v) with a circle of the bundle. The corresponding images {Y=F(X),Y'=F(X')} define a trapezium inscribed in the circle. Circle (c) is the smallest circle among all circles of the bundle passing through {A,B,X} for X moving on v. The contact point D is the symmetric of C with respect to the diameter of c which is orthogonal to e. The middles of lines YY' move on line CD as X varies on (v). Line CD is the conjugate diameter of the direction common to all YY' (equal to the direction of line (g)).
Thus D can be found geometrically and the hyperbola can be constructed as the conic passing through the five points {A,B,D,A',B'}, where AA', BB' are parallel to (g) and have their middles on CD.
Note that since circle (c) does not depend on the direction (e) the parabolas resulting by varying this direction are all tangent to (c) at a point D easily constructed from (e).
See the file CircleBundleTransformationParabola2.html for the figure of parabola in the case {A,B} coincide.

Transforming lines to parabolas

See Also