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Parametrising a surface

  1. Oct 30, 2008 #1
    Hi there!

    I´ve come upon the following problem: I want to determine the set of all lines of the following surface:

    [tex]x_1^2+x_2^2-x_3^2=1[/tex]

    Here´s my idea: if one could determine one line parametrised by an angle and a radius, one could define the whole surface as rotation of this line over the x_3-axis. One can write every point of the surface as [tex]rcos\phi,rsin\phi,\sqrt{r^2-1}[/tex] where r is supposed to be the vector of any circle, got by a slice, parallel to the x_1x_2-plane

    We then go to the otherside of the x_1x_2-plane, to get (by symmetry) onother circle with radius of the same length. If we define another point on it, it will be of the form [tex]rcos\psi,rsin\psi,-\sqrt{r^2-1}[/tex].

    I claim (by intuition) that the phase difference between the angles will be constant for every line, but I cannot prove it :(

    Can someone please help me, or give me a hint?

    Thanks a lot in advance, Marin
     
  2. jcsd
  3. Oct 31, 2008 #2

    tiny-tim

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    Hi Marin! :smile:

    Hint: find the lines that go through (1,0,0). :wink:
     
  4. Nov 1, 2008 #3
    thanks, tiny-tim, it works :) But waht about my claim "the phase difference between the angles will be constant for every line" ? How can I find out if it is right?
     
  5. Nov 1, 2008 #4

    tiny-tim

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    Hi Marin! :smile:

    I honestly can't see the difficulty …

    just calculate where the line through (1,0,0) meets the two planes z = ±a constant. :wink:
     
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