Loading ...
Sorry, an error occurred while loading the content.

Re: [EMHL] Isotomy

Expand Messages
  • Boutte Gilles
    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
    Message 1 of 4 , Apr 20, 2005
    • 0 Attachment
      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.

      Friendly,

      Gilles

      > 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.
      >Friendly
      >François
      >
    • Francois Rideau
      Dear Gilles Fantastic, just what I need! Thanks François [Non-text portions of this message have been removed]
      Message 2 of 4 , Apr 20, 2005
      • 0 Attachment
        Dear Gilles
        Fantastic, just what I need!
        Thanks
        François


        [Non-text portions of this message have been removed]
      • Eric Danneels
        Dear Gilles, you wrote ... Forgive me my ignorance, but what exactly do you mean by a second order transformation ? Does it mean that s(s(P)) = P (sometimes
        Message 3 of 4 , Apr 20, 2005
        • 0 Attachment
          Dear Gilles,

          you wrote

          > 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.
          >

          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
        • Boutte Gilles
          Dear Eric, second order transformation is a formal translation of transormation du second ordre . The definition of a such transformation from Ernest
          Message 4 of 4 , Apr 20, 2005
          • 0 Attachment
            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
            >


            [Non-text portions of this message have been removed]
          Your message has been successfully submitted and would be delivered to recipients shortly.