Let F be a projective transformation. Below I work in the set D of points X such that {X, F(X), F(F(X))} are not collinear. D can be empty (f.e. when F is a translation or homothety) but here I assume that D in non-empty. Under this assumption one can correspond to each line L of the plane a conic in the following way:
[1] The image F(L) = L' is also a line. For every point X of L the image F(X) is a point of L' and line LX which passes through {X, F(X)} is tangent to a conic cL.
The conic cL has the following properties:
[2] Let S be the contact point of line L with the parabola cL. Then S' = F(S) is the intersection point of lines {L, L'=F(L)}.
[3] Point F(F(S)) = S'' is a contact point of cL with line L'.
[4] For every point X of the line L line XX', where X'=F(X) (X on line L') is tangent to cL at a point W which is the harmonic conjugate of V with respect to {X,X'}, where V is the intersection point of XX' with line SS''.
[5] Triangle SS'S'' has the conic cL tangent to its sides {S'S, S'S''} at its vertices {S,S''} and is unique with this property. In other words there is no other point R of the same conic having corresponding triangle RR'R'' with R'=F(R), R''=F(F(R)) and lines {R'R, R'R''} tangent to cL at {R,R''} correspondingly.
The first four claims are proved in the same way are proved the corresponding claims for the special case of affinities studied in AffinityGeneratedParabola.html (*). The only difference is in the kind of the conic, which now is a general one and not necessarily a parabola. This follows from the way the conic is generated. One can namely show easily, introducing projective coordinates along lines SS' and S'S'' that the line coordinates between X and F(X) are in a homographic relation x' = (ax+b)/(cx+d). Hence one can apply the Chasles-Steiner argument and conclude that lines XX' envelope a conic (see Chasles_Steiner.html ).
To prove the fifth claim I apply here a modified argument of that used in (*).
In fact, assume that there are two triangles SS'S'' and RR'R'' as required in [5]. By Brianchon's theorem the diagonals of the quadrilateral formed by the tangents {SS', S'S'', RR', R'R''} and the lines joining the contact points pass through a common point O. There results the figure drawn below.
Define then the projectivity G by requiring {S'=G(S), S''=G(S'), R'=G(R), R''=G(R')}.
- Point O is a fixed point of G.
In fact, point Ï, being the intersection of {SR,S'R'} maps to the intersection point of the lines {S'R',S''R''} which is again Ï, hence this is a fixed point of G.
- The polar PO of O with respect to the conic is an invariant line of G.
In fact, consider the intersection point T of SR with the polar PO. The cross ratio (S,R,O,T)=-1 (harmonic) is preserved by G. Besides line SR maps via G to S'R'. If follows that T maps via G to a point T' on PO.
Analogously it is shown that T' maps via G to the intersection point T'' of S''G'' with PO. This completes the argument showing that PO is invariant under G.
- Assuming now that F is a projectivity mapping every point X on SS' to a point X' on S'S'' so that XX' is tangent to the conic and also every point Y on RR' to a point Y' on R'R'' such that YY' is tangent to the conic we come to a contradiction.
In fact, projectivities F and G coincide at the four points {S,S',R,R'} hence they are identical. Taking {X,X',Y,Y'} along a tangent at a point M of the conic we see that this tangent is invariant under F hence its intersection point M' with PO (which is also invariant under F) is a fixed point of PO. It follows that PO consists entirely from fixed points. Hence the contradiction, since OS should coincide with OS' and OS'' which was excluded at the beginning.
Let F be a projectivity. Let also D be the set of the plane for which {×,×',×''} with X'=F(X), X''=F(X') are not collinear. The previous analysis shows that for each × in D there is a conic, which I denote with k(X) (coincides with cL above), tangent at {×,×''} to lines {××', ×''×'}. The analysis shows that the correspondence is 1-1, i.e. to different points in D correspond different conics.
From the fifth property above follows also that no conic in the subset k(D) of all conics can be invariant under the projectivity F. Thus F introduces a permutation F* in k(D) with no fixed points.
![[alogo]](alogo.png){kind=small}
1. Conic generated from a line by a projectivity ![[0_0]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_0_0_0.png)
![[0_1]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_0_0_1.png)
![[0_2]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_0_0_2.png)
![[1_0]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_0_1_0.png)
![[1_1]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_0_1_1.png)
![[1_2]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_0_1_2.png)
![[0_0]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_1_0_0.png)
![[0_1]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_1_0_1.png)
![[0_2]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_1_0_2.png)
![[0_3]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_1_0_3.png)
![[1_0]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_1_1_0.png)
![[1_1]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_1_1_1.png)
![[1_2]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_1_1_2.png)
![[1_3]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_1_1_3.png)
![[2_0]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_1_2_0.png)
![[2_1]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_1_2_1.png)
![[2_2]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_1_2_2.png)
![[2_3]](ProjectivityGeneratedConic_files/ProjectivityGeneratedConic_1_2_3.png)