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[Differential Geometry of Curves] Regular Closed Curve function

  1. Nov 14, 2012 #1
    1. The problem statement, all variables and given/known data

    Let m : [0,L] --> ℝ2 be a C2 regular closed curve parametrized with arc length, and define, for an integer n > 0 and scalar ε > 2

    μ(u) = m(u) + εsin(2nπu/L)Nm(u)

    where Nm is the unit normal to m

    (1) Determine a maximum ε0 such that μ is a closed regular curve for any ε < ε0

    (2) Assume that m is a circle centered at 0 with radius r > 0
    Prove that μ is non intersecting for any ε < r
    Compute its interior area as a function of r and ε

    2. Relevant equations

    See above

    3. The attempt at a solution

    I think μ(u) is like some scalar of a sine function moving around the original curve. Essentially I need to find a large enough amplitude ε such that μ(u) intersects itself in the middle of m(u).

    It also seems that ε0 must be smaller than the radius of the bitangent circle of m(u)
     
    Last edited: Nov 14, 2012
  2. jcsd
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