Such conjugations could be called

[1] The middles F of AC (C = f(A)) are on a parabola (c').

[2] (c') is a parallel translation of (c) along its axis, by a vector (w) that depends only on the distance of the two axes (d,e) and not their particular location.

[3] For fixed (d,e) the area of the parabolic sector defined by AC is independent of the location of A (and C = f(a)).

[4] It follows that if a fixed angle (at B) glides on a parabola (c) so that its legs BA, BC remain parallel to given directions, then side AC (opposite to B) envelopes a translated parabola (c'), the translation being along the axis of c.

In fact, taking the ordinate x of B and computing (using the symmetries) the coordinates of A, C, we get:

x

Then the coordinates of the middle F of segment AC are seen to satisfy:

y

The other claims on tangency and area (which is equal to 2|e-d|

To obtain a feeling of the whole group of affinities (F) leaving invariant the parabola (c) one can think as follows. F are special cases of projectivities leaving invariant (c), in particular F fix the point at infinity of the parabola. Since (c) is projective image of a circle (see ParabolaProjectFromCircle.html for the realization of such a projectivity G), this amounts to the problem of determination of all projectivities of the circle that leave fixed a particular point P

Note that the family of conics generated by (c) and (c') consists of all parabolas resulting by parallel translating (c) along its axis. This family can be written in homogeneous coordinates: k(y

In the file ParabolaHomographies.html I discuss the projectivities leaving invariant the standard parabola (y=x

Parabola.html

ParabolaHomographies.html

Projectivity.html

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