Definition:Crossing (Jordan Curve)
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Definition
Let $P$ be a polygon embedded in $\R^2$.
Let $q \in \R^2 \setminus \partial P$.
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:
- $\map {\operatorname{par} } q := \map N q \bmod 2$.
Sources
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