The following picture illustrates the relation of the homology coefficient of a perspectivity to the transformation by constant cross ratio applied to a conic. Given is a conic c, a point A not on the conic and a real constant k. Then, for every point E on the conic we define a point I, such that the cross ratio (H,E,A,I) = k, where H is the other intersection point of the conic with line EA. Points I are on a conic c', depending on the given data {c,A,k}. In fact, c' is the image c'=F(c) of c under the perspectivity F with center at A, axis the polar CD of A with respect to c and homology coefficient r = (1+k)/(1-k) = (A,E',E,I).
The proof is a trivial calculation, for example using the conventions explained in ProjectiveLine.html . Fixing a point E on c, and introducing the projective coordinates with basis H(1,0), E(0,1), A(1,1), for which E'(-1,1) and assuming that I has coordinates (x,y), we find that the cross ratio (H,E,A,I) = k = x/y and also (A,E',E,I) = r = (1+k)/(1-k). Thus, the condition of constancy of k is equivalent to the constancy of r and this proves the claim.
![[alogo]](alogo.png){kind=small}
Perspectivity and cross ratio ![[0_0]](Perspectivity2_files/Perspectivity2_0_0_0.png)
![[0_1]](Perspectivity2_files/Perspectivity2_0_0_1.png)