How can we parametrize a surface with an angle and radius?

[Marin]

Hi there!

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

$$x_1^2+x_2^2-x_3^2=1$$

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 $$rcos\phi,rsin\phi,\sqrt{r^2-1}$$ 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 $$rcos\psi,rsin\psi,-\sqrt{r^2-1}$$.

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

 

[tiny-tim]

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

$$x_1^2+x_2^2-x_3^2=1$$

Hi Marin!

Hint: find the lines that go through (1,0,0).

 

[Marin]

thanks, tiny-tim, it works :) But what about my claim "the phase difference between the angles will be constant for every line" ? How can I find out if it is right?

 

[tiny-tim]

thanks, tiny-tim, it works :) But what about my claim "the phase difference between the angles will be constant for every line" ? How can I find out if it is right?

Hi Marin!

I honestly can't see the difficulty …

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