ParabolaHomographies

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 I_b represents the harmonic perspectivity with respect to the point of the x-axis with ordinate x_b = -1/b and axis the polar of this point with respect to the conic. In other words x_b is the Fregier point of the involution preserving the conic and the polar is the homography axis.
Matrix S_a 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 {I_b} preserving the conic and fixing (0,0) do not generate all the group of homographies preserving the conic and fixing (0,0).

Parabola homographies

See Also