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Re: [EMHL] Re: Locus

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  • Francois Rideau
    Dear Jeff Sorry, Cte means constant . So you look at events (m,t(0)) in H where t(0) is some fixed time that I have named Cte in my previous post.. As H is
    Message 1 of 240 , Apr 1, 2007
      Dear Jeff
      Sorry, Cte means "constant".
      So you look at events (m,t(0)) in H where t(0) is some fixed time that I
      have named "Cte" in my previous post.. As H is an hyperboloid, these events
      are on a conic, intersection of H in W = P x T with the plane t = t(0).
      So you see why the Newtonian spacetime W is useful, allowing some synthetic
      proofs in this difficult theory, even if it has no physical existence.
      In fact you have a group G(alilean) acting not only on the set W of events
      but also on the set of world lines, that's just the old theory of Galilean
      relativity.
      Of course, the (choo-choo) theory is the study of the subgroup of G leaving
      hyperboloid H globally invariant. This group G' is known for a long time as
      H is a ruled quadric.
      G' is itself the union of 2 distinct subsets G" and G'" where G" is the
      subgroup of G' leaving globally invariant each of the 2 set of generators of
      H and G'" the subset (G'" is not a subgroup) of G' swapping these 2 distinct
      sets of generators, that is to say G" is a subgroup of G' of index 2.

      Friendly

      Francois




      On 4/1/07, Jeff Brooks <trigeom@...> wrote:
      >
      > Dear Francois,
      >
      > what is Cte?
      >
      > > I want to give a name to theses conicsbut I don't dare choose!
      > > Geometrically they are the projections along time axis on plane P of
      > > the sections of the hyperboloid H by the planes t = Cte.
      >
      > Sincerely, Jeff
      >
      >
      >


      [Non-text portions of this message have been removed]
    • Antreas Hatzipolakis
      [APH]: Let ABC be a triangle. A line L passing through A intersects the circle with diameter AB again at Ab and the circle with diameter AC again at Ac.
      Message 240 of 240 , Feb 16

        [APH]:

         

        Let ABC be a triangle.

        A line L passing through A intersects the circle with diameter AB again at Ab and the circle with diameter AC again at Ac.

        [Equivalently: Let Ab, Ac be the orthogonal projections of B, C on L, resp.]
        Let A* be the intersection of BAc and CAb.

        Which is the locus of A* as L moves around A?


        [César Lozada]:


        Parametric trilinear equation:

         

        1/u(t) = a*((b^2+c^2-a^2)^2-4*b^2*c^2*c os(2*t)^2)/(2*S)

         

        1/v(t) = 2*(cos(2*t)*c-b)*S - c*sin(2*t)*(a^2+3*b^2-2*cos(2* t)*b*c-c^2)

         

        1/w(t) = 2*(cos(2*t)*b-c)*S + b*sin(2*t)*(a^2+3*c^2-2*cos(2* t)*b*c-b^2)

         

        Regards,

        César Lozada

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