Finding the Total Curvature of Plane Curves I'm trying to find the total curvature (or equivalently, rotation index, winding number etc.) of a plane curve (closed plane curves) given by
$$\gamma(t)=(\cos(t),\sin(nt)), 0\le t\le 2\pi$$for each positive integer $n$.
Looking at the image of thess curves makes me believe that the answer is $0$ when $n$ is even and $2\pi$ when $n$ is odd, but how do I prove this? Calculating the integral of the curvature is extremely complicated!
I learned in my class that total curvature is invariant under regular homotopy, but how do I construct such a homotopy?




I am assuming that the index is the rotation index.
This said, there are much easier ways to compute the index. One way is to compute all zeros of the two components of the derivative of the curve (trivial in the above example); each zero represents a time at which the tangent points towards a cardinal direction; in between zeros, the tangent is constrained to point in one quadrant, so the index does not change; by keeping track of how the tangent moves across quadrants in the plane, you can easily compute the index.
