orientation

Let $\alpha$ be a rectifiable, Jordan curve in ${ℝ}^2$ and $z_0$ be a point in ${ℝ}^2-\mathrm{Im}(\alpha )$ and let $\alpha$ have a winding number $W[\alpha :z_0]$. Then $W[\alpha :z_0]=\pm 1$; all points inside $\alpha$ will have the same index and we define the orientation of a Jordan curve $\alpha$ by saying that $\alpha$ is positively oriented if the index of every point in $\alpha$ is $+1$ and negatively oriented if it is $-1$.