Finding a parametrization of a curve from cartesian equations

Question

I'm in need of some help to address this problem.

Let $C$ be a curve given by two equations:

$x^2+y^2-z^2-1=0$,

$x^2-y^2-z^2-1=0$

Express the curve by means of parametric equations.

Any ideas on how to work on it?

Answer 1

Subtract the two equations. We get $y=0$

Plug in the first $$x^2-z^2=1$$ A parametrization is $$(x=\cosh t,y=0,z=\sinh t); (x=-\cosh t , y=0, z=-\sinh t)$$ In the image below the two surfaces and their intersection.

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Answer 2

If you take the first restriction and plug it into the second, you get

$$ x^2-y^2-(x^2+y^2-1) - 1 =0 \Leftrightarrow -2y^2 = 0 \Leftrightarrow y=0 $$

So, any point on the curve satisfies $y=0$ and $x^2-z^2 = 1$ and you can consider two separate parametrizations:

$$ (\sqrt{t^2+1}, 0, t), t\in \mathbb{R} $$

$$ (- \sqrt{t^2+1}, 0, t), t\in \mathbb{R} $$