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Show reparameterized curvature equals curvature up to a sign

  1. Feb 9, 2017 #1
    1. The problem statement, all variables and given/known data
    Let γ: I → ℝ2 be a smooth regular curve and let λ = γ ο φ with φ: Iλ → I be a reparameterisation of γ. Show, by using the general formula for curvature of a regular curve that κλ = ±κ ο Φ where the ± depends on whether φ is orientation preserving (+) or reversing (-).

    2. Relevant equations

    κ = Det{γ',γ''}/{absval(γ')}3

    3. The attempt at a solution
    So, the way I would approach this problem I guess is by first taking the curvature of γ...
    - Though the problem I immediately run into, and did on another problem I just posted, is whether this should be done in general or for a specific case of γ and λ
    + i.e., I would parameterize γ with t perhaps by <acos(t), bsin(t)> then take φ to be something orientation reversing (maybe like φ = -t) and again as something orientation preserving (like maybe φ = t2?) and then show that the curvature of each λ± equals the curvature of the original curve (up to a sign) reparameterized with Φ.
    + Another question is, why are we using k composed with Φ for the reparameterised curvature when the reparamterization is done by composing with the function φ? Wouldn't the curvature then be something like κλ = ±κ ο φ?

    Any help or guidance is extremely appreciated I'm having a bit of trouble grasping how to start and where to go with this.
  2. jcsd
  3. Feb 16, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
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