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Curvature of a space curve

  1. Jul 15, 2009 #1
    1. The problem statement, all variables and given/known data

    Let [tex]\underline{r}[/tex] be a regular parameterisation of a space curve [tex]C \subset R^{3}[/tex]. Prove that

    [tex]\kappa=\frac{\left\|\underline{\dot{r}}\times\underline{\ddot{r}}\right\|}{\left\|\underline{\dot{r}}\right\|^{3}}[/tex] .

    3. The attempt at a solution

    We have

    [tex]t(u)=\frac{\frac{dr}{du}}{\left\|\frac{dr}{du}\right\|}[/tex]

    so differentiating both sides wrt u we obtain

    [tex]\frac{dt}{du}=\frac{\frac{d^{2}r}{du^{2}}}{\left\|\frac{dr}{du}\right\|}+\frac{dr}{du}\frac{d}{du}(\frac{1}{\left\|\frac{dr}{du}\right\|})[/tex].

    Since

    [tex]\frac{dt}{du}=\kappa\underline{n}[/tex]

    this gets me the curavture in terms of the desired bits (with n too) but I can't seem to get it to the desired result :\

    Thanks for your help!
     
  2. jcsd
  3. Jul 15, 2009 #2
    Last edited by a moderator: Apr 24, 2017
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