How many straight lines are on a hyperboloid of one sheet (using vectors)

Question

Hi I think this problem is beyond my knowledge. How many straight lines are contained on set H? and why? So far I only know that H represents hyperboloid of one sheet

$H=${$(x,y,z)/x^2+y^2-z^2=1)$}

I will be very thakful if anyone can give me some idea to get this solve

Answer 1

An infinity: it can be defined as the surface generated by the rotation of a straight line around an axis which is not coplanar with the straight line.

It is also the set of straight lines which cut $3$ given lines which are non-coplanar and not all parallel to the same plane.

Answer 2

For each value of the parameter $\lambda\in \mathbb R$ the following two lines lie on $H$: $$L_\lambda:x=\cos\lambda -t\sin \lambda,\quad y=\sin\lambda +t\cos \lambda, \quad z=t \\ M_\lambda:x=\cos\lambda -t\sin \lambda,\quad y=\sin\lambda +t\cos \lambda, \quad z=-t $$ And there are no other lines on $H$.