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Math Help - Differential Geometry Help Needed!

  1. #1
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    Exclamation Differential Geometry Help Needed!

    Hey guys... Have a Diff Geometry exam coming up and looking at the past papers this keeps coming up but I cant get my head around it! Can anyone help......

    Sample Question
    Q: The plane through a point on a curve г c R^3 perpendicular to the tangent line is called the normal plane to the curve at the point.

    (a) Show that a curve lies in a sphere if the intersection of all normal planes is non-empty.

    (b) Hence, or otherwise show that the curve parametrized by
    P(ө) = (cos 2ө, -2cosө, sin2ө), ө is an element of [0, 2*Pi] lies in a sphere. Find the centre and radius of the sphere!

    AGH!

    From a Confused Student...
    Last edited by mr fantastic; June 7th 2009 at 10:11 PM. Reason: Edited post title
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  2. #2
    Super Member Rebesques's Avatar
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    a) Let the curve be parametrized by arclength, x=x(s) and let \{t,\eta,b\} be the Frenet-Serret frame. Then the plane \{\eta,b\} always passes through a point P. Express P=\lambda(s)\eta(s)+\mu(s)b(s) and differentiate to obtain \lambda, \mu=0. So \{\eta,b\} always crosses the origin, which means x\in\{\eta,b\} or x(s)=\Lambda(s)\eta(s)+M(s)b(s). Use the Frenet-Serret equations to show that \Lambda, M are constants.


    (Actually, the curvature of x turns out to be constant and the torsion zero, so x is a circle.)


    b) Let the coordinates be x,y,z. We easily see that this circle is the intersection of the circular cylinder x^2+z^2=1 with the parabolic cylinder y^2=2(1+x).
    (...unless i messed up my calcs again)
    Last edited by Rebesques; June 7th 2009 at 09:40 PM.
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