Editing Jordan curve theorem (section)

== Application ==
[[File:Jordan Curve Theorem for Polygons - Proof.svg|thumb| If the initial point ({{math|{{color|red|''p<sub>a</sub>''}}}}) of a [[ray (geometry)|ray]] (in red) lies outside  a simple polygon (region {{math|{{color|red|A}}}}), the number of intersections of the ray and the polygon is [[Even number|even]].<br /> If the initial point ({{math|{{color|green|''p<sub>b</sub>''}}}}) of a ray lies inside the polygon (region {{math|{{color|blue|B}}}}), the number of intersections is [[Odd number|odd.]]]]
{{main|Point in polygon#Ray casting algorithm}}
In [[computational geometry]], the Jordan curve theorem can be used for testing whether a point lies inside or outside a [[simple polygon]].<ref>{{harvs|txt|last=Courant|first=Richard|year=1978}}</ref><ref>{{Cite book|url=https://www.maths.ed.ac.uk/~v1ranick/jordan/cr.pdf|title=1. Jordan curve theorem|date=1978|publisher=University of Edinburgh|location=Edinburg|page=267|chapter=V. Topology}}</ref><ref>{{Cite web|title=PNPOLY - Point Inclusion in Polygon Test - WR Franklin (WRF)|url=https://wrf.ecse.rpi.edu//Research/Short_Notes/pnpoly.html|access-date=2021-07-18|website=wrf.ecse.rpi.edu}}</ref>

From a given point, trace a [[ray (geometry)|ray]] that does not pass through any vertex of the polygon (all rays but a finite number are convenient). Then, compute the number {{mvar|n}} of intersections of the ray with an edge of the polygon. Jordan curve theorem proof implies that the point is inside the polygon if and only if {{mvar|n}} is [[parity (mathematics)|odd]].