## Parabola homographies

Here I study the homographies (or projectivities) leaving invariant a parabola (the standard one y=x2) and fixing a point of it (the origin (0,0)). The equation of the parabola in the projectification of the plane is x2 - y*z = 0. A rational parametrization of it (inverse to a good parametrization) is given through the map (u,v) -->(u*v, u2, v2). A homography of the parabola is represented in this coordinate system with a linear map: (u',v') = (a*u+b*v, c*u+d*v). If the homography fixes (0,0) then coefficient b must be zero. Thus the coordinate representation of the homography becomes (u',v') = (a*u , b*u+c*v). and expressing (u'*v', u'2, v'2) in terms of (u*v, u2, v2) we come to the relations:

There are two particular cases of particular interest: the case of harmonic perspectivities preserving the conic as well as the case of affinities. These are represented respectively with the matrices:

Matrix Ib represents the harmonic perspectivity with respect to the point of the x-axis with ordinate xb = -1/b and axis the polar of this point with respect to the conic. In other words xb is the Fregier point of the involution preserving the conic and the polar is the homography axis.
Matrix Sa represents the general affine transformation, leaving invariant the parabola and the line at infinity, (x' = a*x, y' = a2*y) and fixing point (0,0).
The form of the matrices shows that the involutions {Ib} preserving the conic and fixing (0,0) do not generate all the group of homographies preserving the conic and fixing (0,0).