orientationOrientation2002-08-14 15:10:122002-08-14 15:51:22Definition% this is the default PlanetMath preamble. as your knowledge
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\usepackage{amssymb}
\usepackage{amsmath}
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% define commands hereLet $\alpha$ be a rectifiable, Jordan curve in $\mathbb{R}^{2}$ and $z_{0}$ be a point in $\mathbb{R}^{2} - \operatorname{Im}(\alpha)$ 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 \textbf{orientation} of a Jordan curve $\alpha$ by saying that $\alpha$ is \textbf{positively oriented} if the index of every point in $\alpha$ is $+1$ and \textbf{negatively oriented} if it is $-1$.