[alogo] Parabola from circle through projectivity

One can generate a parabola from a circle through a transformation of the circle-points as follows.
Consider a point B on a circle c(A,|AB|=r), and the diameter DC orthogonal to the radius AB. For every point E of the circle draw line BE intersecting line DC at H. Take G on line BE to be the harmonic conjugate of H with respect to B and E (cross ratio (B,E,H,G) = -1). As E moves on the circle G describes a parabola.
The parabola passes through points D and C, and is represented by the equation y = (r-1)x2 with respect to the coordinate axes with origin at B and parallel to BA and DC. The focal distance p=|FB| of the parabola is p = r/4.

[0_0] [0_1]
[1_0] [1_1]

One can extend the transformation E --> G, by the same recipe, to a transformation f, mapping the plane onto itself. The recipe, defining the transformation, assigns to every point X of the plane point Y = f(X) , such that (B,X,HX,Y) = -1. HX being the intersection point of BX with line CD. On the line tB tangent to the circle at B, the map corresponds to every point with ordinate x the point with ordinate x/2. This is in accordance with the fact that for X on tB the corresponding XH is the point at infinity on this line. This map is a perspectivity (see Perspectivity.html ), fixing the points of line DC and the point B, leaving invariant tB and sending line tB', the tangent at the antipodal B' to B, to the line at infinity. The homology coefficient is easily seen to be 2.
Applying homotheties centered at B we see that all parabolas are homothetic to each-other, the homothety interchanging corresponding circles (c). Projectivity f maps the bundle of circles tangent to tB at B to the family of parabolas tangent at the same line at the same point.
The algebraic aspect of this subject is discussed in Projectivity.html , where it is derived the representation of the transformation (f) with a matrix.

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