AffinityGeneratedParabola

1. Parabola generated from a line by affinity

Let F be an affine transformation (affinity see Affinity.html ). Under certain conditions (see Remark-2 at the bottom) one can correspond to each line L of the plane a parabola in the following way:
[1] The image F(L) = L' is also a line. For every point X of L the image F(X) is a point of L' and line L_X which is the line passing through {X, F(X)} is tangent to a parabola c_L.
Parabola c_L has the following properties:
[2] Let S be the contact point of line L with the parabola c_L. Then S' = F(S) is the intersection point of lines {L, L'=F(L)}.
[3] Point F(F(S)) = S'' is a contact point of c_L with line L'.
[4] For every point X of the line L line XX', where X'=F(X) (X on line L') is tangent to c_L at a point W which is the harmonic conjugate of V with respect to {X,X'}, where V is the intersection point of XX' with line SS''. Especially the middle T' of SS' maps to F(T') = T'' which is the middle of SS'' and line т'т'' is tangent to the parabola at its middle T.
[5] Triangle SS'S'' has parabola c_L tangent to its sides {S'S, S'S''} at its vertices {S,S''} and is unique with this property. In other words there is no other point R of the same parabola having corresponding triangle RR'R'' with R'=F(R), R''=F(F(R)) and lines {R'R, R'R''} tangent to c_L at {R,R''} correspondingly.

To prove [1] take appropriate points on L starting with the intersection point S' of lines L and L'. Because of the invertibility of F, there must be point S on L with F(S) = S'. Let S''=F(S')=F(F(S)).
Taking the coordinate origin at S' and axes correspondingly the lines {L,L'} define x = S'X and y = S'X'. Using the property of affinities to preserve ratios along lines we see easily that a relation of the form y = ax+b, with constants {a,b} is valid. Thus [1] results by applying the property examined in the file ThalesParabola.html .
Properties [2] and [3] result easily from the definition of the parabola, as an envelope, by which its points are limiting points of intersections of two neighbour lines, as for example lines SS' and XX', when XX' tends to coincide with SS'. Obviously this limit point is S. Analogously we see that S''=F(S') is a parabola point.
Claim [4] results from the duality of pole-polar. Obviously SS'' is the polar of S' with respect to the parabola and passes through V, hence the polar of V passes also from S'. The claim results by projecting the cross-ratio (X,X',V,W) on XX' from S' on line SS''.
For [5] there is possibly a simpler proof. Here I apply Brianchon's theorem (see Brianchon.html ). To simplify the figure I start using an ellipse.
Assume that there are two triangles SS'S'' and RR'R'' as required in [5]. By the aforementioned theorem the diagonals of the quadrilateral formed by the tangents {SS', S'S'', RR', R'R''} and the lines joining the contact points pass through a common point O.

Point о, being the intersection of {SR,S'R'} maps to the intersection point of the lines {S'R',S''R''} which is again о, hence this is a fixed point of F. Then the ratios SO/OR = S'O/OR' = S''O/OR'' show that lines {SS'', RR''} are parallel. Thus SS''RR'' is a trapezium and in the case c_L is a parabola line S'R' joins the middles of the parallel sides and is parallel to the axis of the parabola.
The rest of the proof follows from properties of the parabola circumscribing trapezium SS''RR'' (see ParabolaCircumscribingTrapezium.html ). We can define projectivity F' by requiring from it to map {F'(S) = S', F'(S') = S'', F'(R) = R', F'(R') = R''}. Projectivity F' coincides then with affinity F (considered as a special projectivity) at four points {S,S',R,R'} hence, by the general properties of projectivities, maps F and F' are identical. But in the above reference it is proved that F' can never be an affinity.

2. Correspondence of a parabola to a point

Let F be an affinity. Let also D be the set of the plane for which {в,в',в''} with X'=F(X), X''=F(X') are not collinear. The previous analysis shows that for each в of D there is a parabola tangent at {в,в''} to lines {вв', в''в'}. The analysis shows that the correspondence is 1-1, i.e. to different points correspond different parabolas.
Remark-1: There are affinities (like reflexions, translations, homotheties) for which D is empty.
Remark-2: In fact the previous arguments apply to points X lying in the set D.
Remark-3: The parabola c_L is a special case of a conic generated by Chasles-Steiner method (see Chasles_Steiner.html ).
File ProjectivityGeneratedConic.html handles a generalization of the subject by which the affinity is replaced by a projectivity leading to more general conics than parabolas.

1. Parabola generated from a line by affinity

2. Correspondence of a parabola to a point

See Also