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Math Help - Show that this curve is contained in a plane

  1. #1
    Senior Member Educated's Avatar
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    Show that this curve is contained in a plane

    Show that a curve \gamma such that \dddot{\gamma} = 0 everywhere is contained in a plane
    So of course the first thing I did was integrated it a few times to get:

    \gamma (t) = (a_1 t^2 + b_1 t + c_1 ,  a_2 t^2 + b_2 t + c_2, a_3 t^2 + b_3 t + c_3)

    where the a, b and c's are constants.

    Now how do I continue from there?

    EDIT:

    Actually I think I have it,

    The parametric equation of a plane is \gamma (t) = (a_1 t + b_1 u + c_1 ,  a_2 t + b_2 u + c_2, a_3 t + b_3 u + c_3)
    where t and u are the varying parameters. In the above integrated equation, there's 2 varying parameters, t and t^2 so it almost fits the equation of a plane, but in this case u is restricted to u=t^2
    Last edited by Educated; July 14th 2014 at 08:24 PM.
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  2. #2
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    Re: Show that this curve is contained in a plane

    Or you could just say that if the third derivative of your curve is 0 everywhere, then the function must be a quadratic, which has a parabola as its graph. Clearly as a 2 dimensional curve it must lie on a plane...
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  3. #3
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    Re: Show that this curve is contained in a plane

    If you have taken, or are taking, differential geometry (this looks like an elementary differential geometry problem to me) you could just show that if the third derivative of the position vector is the 0 vector then the torsion is 0.

    (If t is the unit tangent vector at any point on the curve, then t' is perpendicular to the curve. It length \kappa= |t'| is the "curvature". The unit normal is n= \frac{t'}{\kappa}. The 'binormal' is their cross product: b= t\times n. The torsion is the length of the derivative of b: \tau= |b'|.)

    The fact that the torsion is a multiple of the third derivative:
    \tau= \frac{(\gamma'\times \gamma'')\cdot \gamma'''}{||\gamma'\cdot\gamma''||}
    (http://en.wikipedia.org/wiki/Torsion_of_a_curve)
    makes this problem easy.
    Last edited by HallsofIvy; July 15th 2014 at 06:50 AM.
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