# Definition:Crossing (Jordan Curve)

## 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 $q$.

Then $\LL \cap \partial P$ consists of a finite number of line segments, where $\partial P$ denotes the boundary of $P$.

As two adjacent sides in $P$ do not form a straight angle by the definition of polygon, each line segment is either a single point or an entire side of $P$.

Each of these line segments is called a crossing if and only if the line segment is one of these:

a single point which is not a vertex of $P$
a single vertex of $P$, and its adjacent sides lie on opposite sides of $\LL$
a side $S$ of $P$, and the two sides adjacent to $S$ lie on opposite sides of $\LL$.

### Parity

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:

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