So if we have a simple closed curve $a$, then the following formula holds:$$ \frac{1}{2 \cdot \pi} \int_0^S \kappa(t) ||a'(t)|| dt = \pm 1 $$ ( turning number)
I know that we have use the following somehow:
$\star^1$ : Let $a$ be a regular curve, $C^2$-closed, then $\frac{1}{2 \cdot \pi} \int_0^S \kappa(t) ||a'(t)|| dt = n_T$. ( turning number)
$\star^2$ : Let $a$ be a regular curve, simple $C^1$-closed, then $|n_T|$=1.
So where is my problem? It seems so that the claim above follows directly from $\star^1$ and $\star^2$, but I have one problem: In the claim I only have that $a$ is simple closed curve. Moreover in $\star^1$ we have a $C^2$ closed curve and in $\star^2$ we have a simple $C^1$ closed curve. Can somebody help me?