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The parity of transverse ray crossings with a polygon is locally constant on its complement
Statement
For a polygon (Polygonal arcs and polygons as non-self-intersecting finite unions of line segments in ) and , count the intersections modulo two of any general-position ray supplied by Every point off a polygon admits a ray meeting it transversely in finitely many nonvertex points. This parity is independent of the chosen general-position ray and is constant throughout some open neighbourhood of in the complement. Consequently it is constant on every region of (Regions of the complement of a planar set and their frontiers). The elementary alternation at successive transverse crossings follows the convention of The even and odd index maps and the alternating sequence: strictly increasing with their disjoint union, and the unique with , , which satisfies , and .
Facts & Assumptions
Given: A polygon and a point .
Every point off a polygon admits a ray meeting it transversely in finitely many nonvertex points (Every point off a polygon admits a ray meeting it transversely in finitely many nonvertex points).
Proof
Fix a general-position ray from . The finite intersection points have positive distance from every polygon vertex and from every nonincident edge; transversality also supplies a positive angle at each crossing. Taking the minimum of finitely many positive tolerances gives a ball about in which parallel translated rays retain exactly these crossings.
Rotate one general-position ray continuously to another, avoiding the finitely many exceptional directions except at isolated parameters. Crossing an edge tangentially creates or destroys two intersections, while passing a polygon vertex transfers the intersection from one incident edge to the other or changes the count by two. Thus the count modulo two never changes, so parity is independent of the chosen ray.
Steps 1.1 and 1.2 make parity locally constant on the complement. A locally constant map to the discrete set is constant on each connected component, hence on each complementary region.
Depends on
- Every point off a polygon admits a ray meeting it transversely in finitely many nonvertex points
- Polygonal arcs and polygons as non-self-intersecting finite unions of line segments in $\mathbb R^2$
- Regions of the complement of a planar set and their frontiers
- The even and odd index maps and the alternating sequence: strictly increasing $e, o$ with $\mathbb{N}$ their disjoint union, and the unique $(s_k)$ with $s_0 = 1$, $s_{\sigma(k)} = -s_k$, which satisfies $|s_k| = 1$, $s \circ e \equiv 1$ and $s \circ o \equiv -1$
Used by
Dependency tree · two levels
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Sources
- R. Diestel, Graph Theory, 6th ed., Chapter 4, Section 4.1 (standard reference, not scraped)