Thanks!
It occurs to me to ask: Is there a parametrization of the one-sheeted hyperboloid that makes it manifest that it's a ruled surface?
I believe the following is a valid parametrisation, for much the same reason as the hyperbolic slot demonstration works: x = p cos(theta) + sin(theta), y = p sin(theta) - cos(theta), z = p, where p is a real number; theta is in the interval (-pi,pi]. When theta is constant, varying p corresponds to motion along a straight line. Substituting these expressions into x^2 + y^2 = z^2 + 1 confirms that this parametrisation does indeed describe the hyperboloid.
And: Is there a parametrization that makes it manifest that it's a doubly-ruled surface?
Yes, if there are twelve constants, {a,b,c,d,e,f,g,h,i,j,k,l}, such that: x = apq + bp + cq + d, y = epq + fp + gq + h, z = ipq + jp + kq + l, x^2 + y^2 = z^2 + 1, for all values of {p,q} real. Equating parts, these constraints collapse into nine quadratic equations in twelve variables, which (when solved) will give a valid parametrisation. (Proof: if either p or q is varied, x, y and z will all change linearly.) Sincerely, Adam P. Goucher