30A99-Orientation.tex

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\pmcanonicalname{Orientation}
\pmcreated{2013-03-22 12:56:09}
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\pmtitle{orientation}
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\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{30A99}
\pmrelated{SensePreservingMapping}

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\begin{document}
Let $\alpha$ 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 $-1$.
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