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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 :)
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 :)
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