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Poincare model help

  1. Nov 9, 2008 #1
    Incidence Postualte I-1 holes for the Poincare Model: Every two points of E lie on exactly one L-Line.
    Prove: Given any two points P and Q inside the unit circle C, there exists a unique L-line l containing them. (this will require the use of analytic geometry.)

    [​IMG]

    L-lines:arcs of circles perpendicular to the unit circle in S and the diameter of S.

    How would i solve this? I know that i need to prove and P,Q are not equal to the orgin and that either one is at the orgin, but how?

    Thanks for the help!!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 9, 2008 #2

    HallsofIvy

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    You don't prove that neither P nor Q is at the origin- there is no reason for that to be true. However, a "hyperbolic line" in the Poincare circle model is either
    1) A Euclidean line through the center of the bounding circle (here the origin) or
    2) A circle orthogonal to the bounding circle.

    IF is P or Q is the origin, then it is easy: the Euclidean line from P to Q is a hyperbolic line. You should also be able to prove, with Analytic geometry, that a circle passing through the origin cannot be orthogonal to the unit circle.

    If neither P nor Q are at the origin, then you need to show that there is exactly one circle through both P and Q that is orthogonal to the unit circle. How you would do that, I can't say because I don't know what concepts you know that you can use. I myself would use the fact that the inverse points to P and Q must also be on that circle. Do you know what an "inverse point" in this sense is?
     
  4. Nov 9, 2008 #3
    Do you mean like P' and Q' are inverse points?
     
  5. Nov 9, 2008 #4
     
  6. Nov 9, 2008 #5

    HallsofIvy

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    I have no idea because I have no idea what YOU mean by P' and Q'.
     
  7. Nov 9, 2008 #6

    HallsofIvy

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