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Curvature and Osculating circle

  1. Jan 26, 2009 #1

    bfr

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    1. The problem statement, all variables and given/known data

    Find a formula for the curvature of the curve:

    x=(e^t + e^(-t))/2
    y=(e^t - e^(-t))/2

    Write an equation of the osculating circle when t=0.

    2. Relevant equations

    curvature=|x'y'' - x''y'|/(x'^2 + y'^2)^(3/2)

    3. The attempt at a solution

    First, wouldn't the formula for curvature be:
    sqrt(2)
    ---------
    (e^(2t) + e^(-2t))/2 * sqrt(e^(2t) + e^(-2t))

    ? But for some reason, my teacher marked a big "X" across this and wrote question marks.

    EDIT: Nevermind about the first part of the question - I just should have simplified it more.
    Now, for the equation of the osculating circle when t=0, I'd get the center of the circle by doing: <x_center,y_center>=<x(0)-K*y'(0)/sqrt(x'(0)^2+y'(0)^2),y(0)+K*x'(0)/sqrt(x'(0)^2+y'(0)^2)>

    where K=curvature?
     
    Last edited: Jan 26, 2009
  2. jcsd
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