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.
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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Parabola generation using a circle ![[0_0]](ArtztIsosceles3_files/ArtztIsosceles3_0_0_0.png)
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