Definition:Crossing (Jordan Curve)/Parity

Definition

Let $P$ be a polygon embedded in $\R^2$.

Let $q \in \R^2 \setminus \partial P$.

Let $\mathbf v \in R^2 \setminus \set {\mathbf 0}$ be a non-zero vector.

Let $\LL = \set {q + s \mathbf v: s \in \R_{\ge 0} }$ be a ray with start point $q$.

Let $\map N q$ be the number of crossings between $\LL$ and the boundary $\partial P$ of $P$.

Then the parity of $q$ is defined as:

$\map {\operatorname{par} } q := \map N q \bmod 2$.

Also see

• Jordan Polygon Parity Lemma: $\map {\operatorname{par} } q$ is independent of the choice of $v$

Sources

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• Jeff Ericsson: Computational Topology