What is Winding the line on the circle?

To each real number $t$, associate the point $M(t)$ of $\mathcal C$ obtained by walking a distance $\abs t$ along the circle from $I$, counterclockwise if $t \geq 0$, clockwise if $t < 0$. Since the circumference is $2\pi$, the numbers $t$ and $t + 2k\pi$ ($k \in \Z$) reach the same point: $M(t + 2k\pi) = M(t)$.