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Arc length parametrisation question (error in notes?)

  1. Oct 5, 2013 #1
    This is from my course notes

    http://img28.imageshack.us/img28/2630/ckyl.jpg [Broken]

    In line 3, there's the integral [tex]\int_0^t ||y'(s)||ds[/tex] which represents the length of the curve as a function of t (which I am thinking of as time). Here, I think s is a dummy variable for time.

    The equation in line 4, however, says that s is an element of [0, |C|] which seems to imply that s iis a length. Here, it makes sense, because [itex]\sigma[/itex] maps time to length, and so [itex]\sigma^{-1}[/itex] maps length back to time, which goes into the function y and y(t) is spit out (the position vector function of the curve.

    But back to the integrand, ||y'(s)|| where the variable s is a length makes no sense to me because y is meant to be function of time, not length. Is there something I'm misunderstanding here?

    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Oct 5, 2013 #2


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    t is the label on the x-axis; s is the arc length. The definitions used here have s=0 when t=0, and that s=|C| when t=1.
  4. Oct 5, 2013 #3
    Isn't t a time parameter which is not represented on the coordinate axes?
  5. Oct 8, 2013 #4


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    - if you integrate the speed vector of a parameterized curve whose derivative is never zero( which guarantees that the curve can not back up on itself) the the integral equals the length of the curve.

    - y is just the position in Euclidean space of the curve. It can be a function of many different parameters. s is the parameter whose value equals the length of the curve that has been traversed up to that point.

    - There is no real concept of time here as in physics. t is just a parameter. s is another parameter.
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