Differential Gemoetry

1. Jan 31, 2009

latentcorpse

Describe the one sheeted hyperboloid as a ruled surface, that is find vector functions $\vec{z},\vec{p}:\mathbf{R} \rightarrow \mathbf{R}^{3}$ such that

$\vec{x}(u,\nu)=\vec{z}(u)+\nu \vec{p}(u)$ parameterises the hyperboloid.

Hint:Let $\vec{z}(u)$ parameterise the circle $x^2+y^2=1$in the $z=0$ plane.

So far I've established \vec{z}(u)= \left[ \begin {array}{c} \cos \left( u \right) \\\noalign{\medskip} \sin \left( u \right) \\\noalign{\medskip}0\end {array} \right] which is pretty obvious. Any ideas on what to do next? Thanks in advance