(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let m : [0,L] --> ℝ^{2}be a C^{2}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)N_{m}(u)

where N_{m}is the unit normal to m

(1) Determine a maximum ε_{0}such that μ is a closed regular curve for any ε < ε_{0}

(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 ε_{0}must be smaller than the radius of the bitangent circle of m(u)

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# Homework Help: [Differential Geometry of Curves] Regular Closed Curve function

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