[Differential Geometry of Curves] Regular Closed Curve function

[cheersdup]

Homework Statement

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)N_m(u)

where N_m 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

Homework Equations

See above

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)

 

[lippyka]

because if it was larger then μ(u) would intersect itself in the middle of m(u).For part (2) I'm not sure how to approach it. I'm guessing I have to use some trigonometry but I'm not sure what.