ParabolaSymmetries

Parabola symmetries

Here I study the affine symmetries of a parabola. The usually discussed ones are the conjugations along parallel directions (A --> B, as AB moves parallel to itself). The middles D of two conjugate points A, B moving along an axis parallel to the parabola axis. Here the parabola is the standard one, represented by the graph of y = x².
Such conjugations could be called affine reflexions since they imitate the reflexion prototype, by reflecting through a beam making a fixed angle with the mirror. The composition of two such reflexions, with mirrors at x=d and x=e, could be named affine rotation. In the figure below, it is the map f (A --> C). f has the following properties:
[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_A = 2d - x, y_A = (2d - x)², x_C = 2e - x, y_C = (2e - x)².
Then the coordinates of the middle F of segment AC are seen to satisfy:
y_F = x_F² + (e-d)² (thus, |w| = |e-d|²).
The other claims on tangency and area (which is equal to 2|e-d|³/3) are proved easily using the above calculations.
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 P₀ of it (assuming that P₀ is projected by G to the point at infinity of the parabola).
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(y²-xz) + k'(z²) = 0, showing that it is also generated by (c) and the line at infinity (z=0) as a degenerate conic (double line).
In the file ParabolaHomographies.html I discuss the projectivities leaving invariant the standard parabola (y=x²) and fixing a point of it (0,0).

Parabola symmetries

See Also