how to find parametrization for an intersection of a plane and one sheet hyperboloid

Question

I need to find a parametrization for intersection of a plane and one sheet hyperboloid.

one sheet hyperboloid equation: $x^2+y^2-z^2=1$

plane equation: $x-1=0$

I don't know how to parametrize the intersection, but I do know that it is an "X" shape.

From the equations I get:

$x=1$

$y^2=z^2$

I tried many combinations like:

$r(t)=(x(t),y(t),z(t))=(1,|t|,t)$ or $(1,|t|,|t|)$

but no matter what I tried I'm not getting the wanted shape. I added a picture that shows the intersection of the hyperboloid and the plane; it's the black "X".

Answer 1

First parametrize the hyperboloid putting $$x = \cosh u \cos v, \quad y = \cosh u \sin v, \quad z = \sinh u $$Now force $x=1$, meaning $\cosh u \cos v = 1$. Write, say, $v = \arccos({\rm sech}\, u)$. Then one parametrization can by $$x = 1, \quad y = \cosh u \sin(\arccos({\rm sech}\, u)), \quad z = \sinh u.$$You can simplify that $y$ coordinate if you want, though.