# [Differential Geometry of Curves] Regular Closed Curve function

1. Nov 14, 2012

### cheersdup

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