Definition:Crossing (Jordan Curve)

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Definition

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

Let $q \in \R^2 \setminus \partial P$, and let $\mathbf v \in R^2 \setminus \left\{ {\mathbf 0}\right\}$ be a non-zero vector.

Let $\mathcal L = \left\{ {q + s \mathbf v: s \in \R_{\ge 0} }\right\}$ be a ray with start point $q$.


Then $\mathcal L \cap \partial P$ 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 iff 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 $\mathcal L$.
  • a side $S$ of $P$, and the two sides adjacent to $S$ lie on opposite sides of $\mathcal L$.


Parity

Let $N \left({q}\right)$ be the number of crossings between the ray $\mathcal L$ and the boundary $\partial P$ of the polygon $P$.

Then the parity of $q$ is defined as:

$\operatorname{par} \left({q}\right) := N \left({q}\right) \bmod 2$.

It follows from the Jordan Polygon Parity Lemma that $\operatorname{par} \left({q}\right)$ is independent of the choice of $v$.


Sources

Jeff Ericsson: Computational Topology