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Compute Frenet apparatus (differential geometry)

  1. Feb 4, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the Frenet apparatus for the curve [tex]\alpha (t) = (at, bt^2, ct^3)[/tex], where [tex] abc \neq 0[/tex].

    2. Relevant equations

    The Frenet equations

    3. The attempt at a solution

    The derivative of the curve is the expression for the tangent vector. The second derivative (the first derivative of the tangent) yield the curvature and normal vector:

    [tex] \alpha ' (t) = T(t) = (a, 2bt, 3ct^2) [/tex]

    [tex] T'(t)=(0, 2b, 6ct) = \kappa (t)N(t)[/tex].

    And here's where I got stuck. I don't know how to separate kappa from the calculated expression for T'. I'm studying for an exam tomorrow and this was on the review sheet, but from day one we've only had curves where we assumed unit speed parametrization. Given the problem statement, this is not an assumption I can make here. But to draw kappa out of the expression for T', I have to know that the normal vector N is of unit length. So was I supposed to reparameterize this from the outset? And if so, I am again at a loss as this was not a practice in our course thus far. I mean, I know that arclength is given by [tex] \int _a ^b \Vert \alpha ' (t) \Vert dt [/tex] but that's as far as I get with that approach.

    Thanks in advance.
     
  2. jcsd
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