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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

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# Parametrising a surface

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