ConicHomography

1. Conic homography

A conic homography (f) is an invertible transformation of the points of a conic (c) onto itself that preserves the cross-ratio of four points on the conic. Such a transformation f can be defined in two ways:
(a) Intrisically, without using the surrounding space but using a good parameterization of the conic (see GoodParametrization.html ).
(b) By restricting on the conic (f=F|c) a projectivity F of the whole plane that preserves the conic (i.e. maps the conic onto itself).
To define the conic homography in the first way fix a good parameterization G_P : c --> R through stereographic projection from a point P of the conic on a fixed line. Then define f by setting its representation f'=G_PfG_P^-1 : R --> R, through a Moebius transformation, i.e. a relation of the form t' = (at+b)/(ct+d), with non-zero determinant ad-bc.
Since this relation is completely determined by giving the values at three different points, we conclude that conic homographies are determined by prescribing the values {A',B',C'} at three points {A,B,C} of the conic (both tripples consist of points of the conic).
To define the conic homography in the second way simply restrict on (c) a projectivity preserving (c) and defined in the whole projective plane. Working in the standard model of the projective plane, and if the conic is given by a symmetric matrix A, so that its equation is x^tAx=0, then such a conic homography (preserving c) is determined through an invertible matrix P satisfying the relation P^tAP = A.

It can be shown (Berger, II p. 178) that the two definitions are equivalent. Below is illustrated such an example of a conic homography of the conic (c). Shown are the two tripples of points {A,B,C} and {A'=f(A),B'=f(B),C'=f(C)} as well as some other points, which are of importance for the study of the conic homography thus defined.
The main ingredient is the homography axis (line (d) below), which is defined by the following property: For a pair of points A, B on the conic and their images A'=F(A), B'=F(B) under the conic homography, lines AB' and BA' intersect on a fixed line (d).

The proof of the assertion on the homography axis is given in ( HomographyAxis.html ). Here I notice that the homography axis (d) is an invariant line of the extension of F on the plane (of the conic homography) and its pole with respect to the conic is a fixed point D of the extension of F on the plane. This follows immediately from the complete quadrilateral shown above. If B'=F(B), join D with B and B' to define B₀ and B₁ on lines BD and B'D respectively. By the assumption that (d) is the homography axis follows that B₁=F(B₀). Thus, line BB₀ is mapped to line B'B₁ and by the preservation of cross ratios D=F(D). The invariance of (d) is also proved with a similar argument.

2. The associated bitangent invariant family

With the previous settings, since the conic (c) and its homography axis (d) are invariant under the (extended) homography F, the same will be true for every conic of the so called bitangent bundle generated by the conic (c) and the doubled line d². Identifying the symbols with their equations, the family can be described by varying parameter k into the equation
c_k = c + kd² = 0 (*).
Theorem-1 Conic homographies are always connected to projectivities F of the plane which leave invariant a whole bitangent family of conics. These conics are described by (*) and include an invariant line (d) and a limiting point D, which is fixed by F. The pair (D, d) is determined by the conic homography and satisfies the pole-polar relation with respect to the conic. In fact, it satisfies this relation with respect to every member c_k of the corresponding invariant family.

3. The two categories of conic homographies, involutive homographies

In case point D is not lying on line d, the picture suggests the distinction of two important categories of conic homographies on a conic (c), the homographies being defined by presrcibing their values on three points of the conic {A,B,C, A'=f(A), B'=f(B), C'=f(C)}:
(I) Lines {AA',BB',CC'} are concurrent.
(II) Lines {AA',BB',CC'} are not concurrent.
Most easily is characterized the first category. In that case D, the fixed point of the extended homography F, coincides with the intersection point of the three lines and the invariant line d remains not only invariant by F but also pointwise fixed.
In fact, the intersection point E of {AB',A'B} is on d and the intersection point G of {AA',BB'} is on the polar of E with respect to the conic. Analogously the intersection point N of {BB',CC'} is on the polar of the intersection point M of {BC',CB'} lying on d. Since, by the duality of pole-polar, the two polars pass through D, if G and N coincide then they are identical with D. In that case lines {DA, DB, DC} are invariant by F, hence their intersections with d are fixed points of F. Thus F has three fixed points on D, consequently every point of it remains fixed under F.
A consequence of it is that every line through D is invariant by F, since it contains two fixed points of F: D and its intersection with d.
A consequence of it is that F has a very simple behaviour similar to the diametric point correspondence of a circle with respect to its center. For each point P on c the image P'=F(P) is the other intersection point of line DP with the conic. A consequence of this is that also F(P')=P, hence F² = I (I : the identity transformation) and F is an involution. Inversely it can easily be shown that an involutive homography of a conic cannot have the fixed point D on its invariant axis without being trivial (the identity) and that it fixes the points of its homography axis and has its lines PP' passing through D. Thus the following is true.

Theorem-2 There are two categories of conic homographies:
(I) The involutive, satisfying F² = I, and
(II) The others, non-involutive.
The first category is characterized by the following facts:
a) The homography axis d remains pointwise fixed under F and there is one more fixed point D not contained in d.
b) Lines PP' (P'=F(P)) pass through the point D.
c) F is a harmonic homology (called also harmonic perspectivity). To each point X it associates a point Y such that the cross-ratio (X,Y,D,H_X)=-1, H_X denoting the intersection of XD with line d.

4. The case D is contained in d

Next picture illustrates the case (in category II) where the fixed point D of F is contained in the homography axis, consequently the invariant line d is tangent to the conic c and the induced in line d homography is represented by a Moebius transformation y=(ax+b)/(cx+d). It shows also some members of the invariant family of conics generated by the conic and the homography axis. All conics of the family are tangent to line d.
Fixing a conic and point D every conic homography of this kind is uniquely determined by a couple of points (A,A') of the conic. In fact, for every other point B of the conic lines A'B and B'A must intersect at a point X on d, hence B' is determined as the other intersection point of line AX. This argument applies more generally and proves the following.
Theorem-3 If the homography axis or the fixed point of the conic homography is known, then the homography is completely determined by a couple of points (A,A') of the conic.

1. Conic homography

2. The associated bitangent invariant family

3. The two categories of conic homographies, involutive homographies

4. The case D is contained in d

See Also