My textbook writes out what the equations of the two one-parameter families of lines that lie on a hyperbolic paraboloid surface are, but I am having trouble figuring out how these would have been determined.
Given a simple hyperbolic paraboloid:
$z = \frac{x^2}{a^2} - \frac{y^2}{b^2}$
How do I determine which family of lines lies entirely on its surface?
On
For the Hyperboloid of One Sheet :
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$$
The two families of ruled lines are given by:
$$ x(u, v) = a(cos(u) − v sin(u)), b(sin(u) + v cos(u), cv) $$
$$x(u, v) = a(cos(u)+v sin(u)), b(sin(u)−v cos(u), −cv) $$
A note on the drawing: The actual construction of the drawing above involves taking discrete values for the variable $v$. I believe I used forty lines from one ruling and forty from the other ruling. It is intended to look like the old-fashioned string models of ruled surfaces. The best reference I can think of to view similar models is Hilbert and Cohn-Vossen , "Geometry and the Imagination".
Write your hyperbolic paraboloid in the form $$H:\quad z=\left({x\over a}+{y\over b}\right)\left({x\over a}-{y\over b}\right)\ .$$ For given $c\in{\mathbb R}$ consider the two planes $$P_c:\quad {x\over a}-{y\over b}=c,\qquad\qquad \hat P_c:\quad z=c\left({x\over a}+{y\over b}\right)\ .$$ The line $$g_c:=P_c\wedge \hat P_c$$ then lies in $H$.
Considering in a similar way for given $d\in{\mathbb R}$ the planes $$Q_d:\quad {x\over a}+{y\over b}=d\qquad\qquad \hat Q_c:\quad z=d\left({x\over a}-{y\over b}\right)$$ gives you the second family of lines on $H$.