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Math Help - Help with curvature proof

  1. #1
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    Help with curvature proof

    If gamma(t) = ( x(t), y(t) ) is a 2 dimensional curve, the curvature
    k(t) at point gamma(t) is this:

    k(t) = A/B

    A = [ (dx/dy)*(d^2y/dt^2) - (d^2x/dt^2)*(dy/dt) ]

    B = [ (dx/dt)^2 + (dy/dt)^2 ]^(3/2)

    can someone help me prove that?

    thank you
    Last edited by tea217; February 25th 2007 at 02:30 AM.
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  2. #2
    Super Member Rebesques's Avatar
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    Remember that the curvature is given by k=dQ/ds, where Q is the angle between the tangent vector (x'(t),y'(t)) and (1,0), and ds the arclength element. Begin by

    k=dQ/ds=(dQ/dt)/(ds/dt),

    and use:

    -> (ds/dt)^2= (x')^2+(y')^2

    but also

    -> Q=Arctan(dy/dx)=Arctan([dy/dt]/[dx/dt]),

    etc.
    Last edited by Rebesques; April 23rd 2007 at 03:40 AM. Reason: stupid me! :D
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