Conic Homography

Given a conic c, a conic homography of c is a bijective map F of c onto c, which by a good parametrization S (see GoodParametrization.html ) is represented through (i.e the map F' = S*F*S-1 is) a line homography (i.e. F' is of the form x'=(ax+b)/(cx+d)).

Since different good parametrizations are related themselves by line homographies, the definition is independent of the particular good parametrization. Since they are (via S) described by homographies they share with them the following fundamental properties:
[1] They preserve the cross ratio of four points on a conic (see CrossRatio.html for its definition).
[2] They are distinguished in involutive (i.e. such that F2=1) and non-involutive..
[3] Involutive/non-involutive are completely determined by 2/3 points of the conic and their images on it.
[4] They can have no/one (the non-involutive)/or two fixed points.
[5] They can be extended to global homographies (or projectivities) of the projective plane.

Involutive homographies are examined in InvolutiveHomography.html . Here we continue with the non-involutive case.
The main characteristic of a conic homography (involutive or not) is its homography axis (see HomographyAxis.html ). This is a line which passes through the fixed points of the homography (if any). Given two points X, Y on the conic and their images X', Y', it coincides with the locus of intersection-points P of lines XY' and X'Y.
In the image below we define a conic homography F by prescribing the images 1', 2', 3' of three points 1, 2, 3 of the conic. The homography axis (pink) is displayed and the intersection point P of lines 13', 1'3 is seen to be on this axis. F1, F2 are the fixed points of F.

The conic (c) and the homography axis (considered as a degenerated double-line-conic) generate a family (I) of conics. Further F can be extended to a projectivity F0 of the whole (projective) plane, which preserves each single member of the family (I). Line (g) is a particular one. It is the pre-image under F0 of the line at infinity i.e. F0(Q) is at infinity for every point Q on line (g). In this example of homography, the family (I) is a bitangent family of conics and (g) is tangent to the unique parabolic member of this family. These matters are discussed in BitangentConics.html .