# 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 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:

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