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Finding Curvature

  1. Jun 24, 2013 #1
    1. The problem statement, all variables and given/known data
    Find the curvature K of the curve, where s is the arc length parameter:

    [itex]\vec{r}(t) = \langle 2 \cos t , 2 \sin t, t \rangle[/itex]




    2. Relevant equations

    [itex]s(t) = \int_a ^t ||\vec{r}'(u)||du[/itex]

    3. The attempt at a solution

    I know I need to find the arc length function, in order to find the curvature function; however, I am unsure as to what I should choose a to be for the lower limit of the integral.
     
    Last edited: Jun 24, 2013
  2. jcsd
  3. Jun 24, 2013 #2

    pasmith

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    You need to use the Frenet-Serret formulae and the chain rule.

    You should not need to perform any integrations.
     
    Last edited: Jun 24, 2013
  4. Jun 24, 2013 #3

    ehild

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    Choose it so that s(0)=0.
     
  5. Jun 24, 2013 #4
    So, the choice is arbitrary? If so, why?
     
  6. Jun 24, 2013 #5

    ehild

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    Why not? Just as you can start time at any instant, you can measure an arc length from any point on a line. Usually the arc length is connected to the trajectory of an object if time is involved, that is why I suggested s(0)=0. But you also can keep "a" in the expression s(t). Solve the problem and you will see that"a" cancels.

    ehild
     
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