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Geodesics by Plane Intersection

  1. Jan 3, 2016 #1
    I want to try and see the intersection between the hyperboloid and the 2-plane giving an ellipse. So far I have the following:

    I'm going to work with ##AdS_3## for simplicity which is the hyperboloid given by the surface (see eqn 10 in above notes for reason) ##X_0^2-X_1^2-X_2^2+X_3^2=L^2##

    If I take the eqn of the 2-plane to be (see Figure 11) ##X_0+X_2=Le^{w/L}## then ##X_0^2+X_2^2=L^2e^{2w/L}-2X_0X_2##

    Substituting for the intersection gives ##(X_0+X_2)^2-X_1^2-2X_2^2+X_3^2=L^2 \quad \Rightarrow L^2 e^{2w/L} -2X_0X_2-X_1^2-2X_2^2+X_3^2=L^2## which I don't recognise as anything to do with an ellipse?

    EDIT: solved :)
     
    Last edited: Jan 3, 2016
  2. jcsd
  3. Jan 4, 2016 #2
    Please show us what you did.
    Also post the figures you reference.

    This is common courtesy towards people that find this through google or forum search.
     
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