[alogo] Parabola generation using a circle

Consider a chord IJ of a parabola c, which is orthogonal to its axis. Let also c' be a circle on the same chord IJ. Consider a point P moving on the parabola c and draw a parallel to its axis intersecting line JI at E. Let Q be the second intersection point of line AE with the circle, where {A,H} are diametral points of the circle on the axis of the parabola. Let also O be the middle of JI and OP the intersection point of PQ with JI. Finally let C be the intersection point of PQ with the axis of the parabola. The computations below are done in the coordinate system with origin at O, line JI as x-axis and line OH as y- axis. The following properties hold.
[1] C is a fixed point on the axis.
[2] The cross-ratio k = (C,OP,P,Q) is constant.
[3] The projectivity fixing points {I,J,C} and mapping G to H is a perspectivity with homology coefficient k and maps the parabola to the circle.

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

The properties follow from the first one, which is an easy calculation. In fact if D(0,d) the center of the circle, E(x,0), then P(x,y) is of the form (parabola) y = s-a*x2, with s/a = OJ2 = R2 - d2. Then EQ*EA = EJ*EI = (OJ-x)*(OJ+x) = OJ2-x2 ... etc.. leading to AC = 2a(R2-d2) and the constancy of C.
From this follows the constancy of k and this implies the third property by the very definition of the perspectivity (see Perspectivity.html ).

Remark This is yet another view of the possible perspectivities which map a circle to a parabola. Some other viewpoints are discussed in ParabolaProjectFromCircle.html as well as in Projectivity.html .

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