[alogo] Perspectivity and cross ratio

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).

[0_0] [0_1]

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.

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