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

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

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

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.html

ParabolaHomographies.html

ParabolaSymmetries.html

Projectivity.html

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