(a) Intrisically, without using the surrounding space but using a

(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

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

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

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 B

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

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 F

(I) The involutive, satisfying F

(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

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.

Harmonic_Perspectivity.html

HomographyAxis.html

HomographyConic3by3.html

Projectivity.html

Perspectivity.html

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