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Thread: Differential Geometry Help Needed!

  1. #1
    May 2008

    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!


    From a Confused Student...
    Last edited by mr fantastic; Jun 7th 2009 at 10:11 PM. Reason: Edited post title
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  2. #2
    Super Member Rebesques's Avatar
    Jul 2005
    My house.
    a) Let the curve be parametrized by arclength, $\displaystyle x=x(s)$ and let $\displaystyle \{t,\eta,b\}$ be the Frenet-Serret frame. Then the plane $\displaystyle \{\eta,b\}$ always passes through a point $\displaystyle P$. Express $\displaystyle P=\lambda(s)\eta(s)+\mu(s)b(s)$ and differentiate to obtain $\displaystyle \lambda, \mu=0$. So $\displaystyle \{\eta,b\}$ always crosses the origin, which means $\displaystyle x\in\{\eta,b\}$ or $\displaystyle x(s)=\Lambda(s)\eta(s)+M(s)b(s)$. Use the Frenet-Serret equations to show that $\displaystyle \Lambda, M$ are constants.

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

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