ParabolaProjectFromCircle

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.

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,H_X,Y) = -1. H_X being the intersection point of BX with line CD. On the line t_B 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 t_B the corresponding X_H 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 t_B and sending line t_B', 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 t_B 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.

Parabola from circle through projectivity

See Also