I am totally confused by the various notions and formulas for Total curvature/Turning Number/Winding Number of plane curves.
I did some calculations for the curve $\alpha(\phi)=(r(\phi),\phi)$ in polar coordinates with $r(\phi)=(2\cos(4\phi))^{1/4}$.
My result until now is contrary to theory and a glance on the graph the value 2.5
Has anyone a different/correct result ?
The total curvature should be $5\pi$. The curve consists of four lobes, each traversed counterclockwise as $\phi$ increases. There is no winding number, per se, as the curve crosses through the origin numerous times. (Note that $\phi\in [-\pi/8,\pi/8]\cup[3\pi/8,5\pi/8]\cup[7\pi/8,9\pi/8]\cup[11\pi/8,13\pi/8]$.)
EDIT: The tangent turns through $\pi/4 + \pi = 5\pi/4$ on each of the four lobes, so the total curvature is in fact $5\pi$. This gives a surprising result because of the singular nature of the curve.
If we imagine smoothing the points of the curve at the origin and having the curve go smoothly from $\phi = \pi/8$ to $3\pi/8$, etc., then the smoothly-varying tangent turns through $-3\pi/4$ each time. Thus, the total curvature of the given curve $-3\pi$ gives us the usual $2\pi$ for a simple closed, smooth curve, traversed counterclockwise. That is, the total curvature of the given curve is, indeed, $5\pi$.