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Differential Gemoetry

  1. Jan 31, 2009 #1
    Describe the one sheeted hyperboloid as a ruled surface, that is find vector functions [itex]\vec{z},\vec{p}:\mathbf{R} \rightarrow \mathbf{R}^{3} [/itex] such that

    [itex]\vec{x}(u,\nu)=\vec{z}(u)+\nu \vec{p}(u)[/itex] parameterises the hyperboloid.

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

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