Re: [EMHL] Isotomy
- Dear friends,
The line (MM') and the conic Gamma have two common points, M' and P.
Let P' = s(P), then the line (M'P') is tangent to Gamma.
It's true for anyone transformation s of second order.
> Dear friends
>One knows that the isotomic image of a line D is a conic Gamma through the
>vertices of the reference triangle ABC.
>Let M on D and M'=s(D) on Gamma, where s is the isotomy map wrt ABC.
>Is there a simple construction of the tangent in M' at Gamma?
>Thanks for your replies.
- Dear Gilles,
> The line (MM') and the conic Gamma have two common points, M' and P.Forgive me my ignorance, but what exactly do you mean by a second order
> Let P' = s(P), then the line (M'P') is tangent to Gamma.
> It's true for anyone transformation s of second order.
Does it mean that s(s(P)) = P (sometimes called a conjugation)
or is it something else ?
Greetings from Bruges
- 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".
>Dear Gilles,[Non-text portions of this message have been removed]
>Forgive me my ignorance, but what exactly do you mean by a second order
>Does it mean that s(s(P)) = P (sometimes called a conjugation)
>or is it something else ?
>Greetings from Bruges