I want to see how a hyperboloid of revolution is generated by its generatrix, i.e. by rotating one (say the one passing point $(1,0,0)$ and $(0,1,1)$) of the skew lines around another (say the z axis).
My trial:
Manipulate[
Show[
ParametricPlot3D[
{Sqrt[t^2 + (t + 1)^2] Cos@θ, Sqrt[t^2 + (t + 1)^2] Sin@θ, 1 + t},
{t, -1, 1}, {θ, 0, β},
BoxRatios -> {1, 1, 1}, PlotRange -> {{-2, 2}, {-2, 2}, {0, 2}}],
ParametricPlot3D[{1 - t, 1 + t, 1 + t}, {t, -1, 1}]
],
{β, 0.1, 2 π}
]
However, the generatrix $(1 - t, 1 + t, 1 + t)$ doesn't match the hyperboloid. So how can I do it correctly?