Dear Eric,

"second order transformation" is a formal translation of "transormation

du second ordre".

The definition of a such transformation from Ernest Duporcq, Premiers

principes de géométrie moderne, Gauthier-Villars, Paris, 1899, pp. 133 ss :

Let c and c' two correlations. For any point M, the line c(M) intersects

the line c(M') at the point M'.

We define, through c and c', a transformation s : M->M', which we call a

quadratic transformation (e.g. any inversion).

For 3 points M_1, M_2, M_3, we have c(M_i) = c'(M_i) so s(Mi) is not

defined. These points are the "poles" of s, the lines d_i = (M_i) are

the singular lines of s.

A "second order transformation" is a particular quadratic

transformation, for wich the singular lines are d_1 = M_2M_3, d_2 =

M_3M_1, d_3 = MM_2 (e.g. iogonality, isotomy). Then s is an involution.

Thus any quadratic transformation is the composition h.s, with h

homography, s "second order transformation".

But, if s is any axial symmetry , then s is an involution but not a

"second order transformation".

Friendly,

Gilles

>Dear Gilles,

>

>

>Forgive me my ignorance, but what exactly do you mean by a second order

>transformation ?

>

>Does it mean that s(s(P)) = P (sometimes called a conjugation)

>or is it something else ?

>

>Greetings from Bruges

>

>Eric

>

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