HomographyAxis

Homography Axis

The context here is a conic (c) and a homography F preserving the conic (see ConicHomography.html ). Such homographies are defined by fixing 3 points A, B, C and their images A', B', C' correspondingly, all six points on the conic. Such homographies define a line (red), called homography axis with the property:
Every pair of points (A, X) defines with the corresponding (F(A), F(X)) the intersection point P of lines [A,F(X)] and [X,F(A)]. The locus of P, is a line the homography axis of F.

By considering the bundles of lines A'(A,B,C,X) and A(A',B',C',X') the arguments of LineHomographyAxis.html transfer verbatim to prove this case. The only difference is in the interpretation of the cross ratio of four points on a conic. But the definition of the cross ratio for points on a conic implies that this is invariant under homographies preserving the conic.
Notice that the proposition is equivalent to Pascal's theorem, since a hexagon defines by its vertices (several) homographies preserving the conic. For each such homography the homography axis coincides with the line resulting from Pascal's theorem.
Notice that the intersection points (if any) V, W of the homography axis with the conic are the fixed points of F on the conic. The intersection point U of the tangents to the conic (if any) at V and W is a third fixed point of the extension of F to the whole projective plane. See the file ConicHomography.html for a discussion of the extension of F on the whole projective plane (it is always possible and can be done in a unique way).
Consider now the case in which the homography axis intersects the conic and denote the equations of the tangents UW by (v=0), UV by (w=0) and the equation of the chord VW by (u=0). The one-parameter family of conics: v*w - k*u² = 0 (k an arbitrary real number) consists of all conics passing through V, W and having there the prescribed lines (w=0, v=0) as tangents. The homographies of type F', defined by the previous recipe, are precisely the ones that preserve every member c' of this family. There is a particularly interesting member of this family, namely the parabola with chord VW and tangent to the lines UV, UW at V and W correspondingly. The pre-image (g) of the line at infinity via F' is a tangent to this parabola. The file BitangentConics.html contains a geometric construction of this parabola, discusses the properties of the family of conics, in particular the role of line (g) associated to the particular homography F (or its extension F').

Homography Axis

See Also