Category:Single Point Characterization of Simple Closed Contour

This category contains pages concerning Single Point Characterization of Simple Closed Contour:

Let $C$ be a simple closed contour in the complex plane $\C$ with parameterization $\gamma: \closedint a b \to \C$.

Let $t_0 \in \openint a b$ such that $\gamma$ is complex-differentiable at $t_0$.

Let $S \in \set {-1,1}$ and $r \in \R_{>0}$ such that:

for all $\epsilon \in \openint 0 r$, we have $\map \gamma {t_0} + \epsilon i S \map {\gamma '}{t_0} \in \Int C$

where $\Int C$ denotes the interior of $C$.

If $S = 1$, then $C$ is positively oriented.

If $S = -1$, then $C$ is negatively oriented.