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  "body": "\n## Statement\n\nLet $C \\subset \\R^2$ be a simple closed polygonal curve.\nThat is, $C$ is a closed polygonal chain that meets itself only where its last side ends at the start of its first side.\nThen $\\R^2 \\setminus C$ has exactly two connected components.\nOne component is bounded and the other is unbounded, and $C$ is the boundary of each of them.\n\n## Remarks\n\nThe proof uses parity of crossings.\nFor a point $x \\notin C$, take a ray from $x$ in a direction that avoids the vertices of $C$, and count how many times it crosses $C$.\nThe parity of this count does not depend on the ray, and it is constant along paths that avoid $C$.\nSo points of even and odd parity lie in different [[components|0V032VWTHD]].\nConversely, two points of the same parity can be joined by a path that runs close to $C$, so there are exactly two components.\nPoints far away have even parity and form the unbounded one.\n\nThis crossing count is also the standard test of whether a point lies inside a polygon.\n\n## Explanations\n\n### L1 human\n\nA closed loop made of straight pieces that never crosses itself splits the plane into an inside and an outside.\nTo go from inside to outside, a path has to cross the loop.\nA point is inside when a ray from it crosses the loop an odd number of times.\n\n### L2 human\n\nThis is the special case of the [[Jordan curve theorem|687RGCHBTP]] for polygonal curves, and its proof is elementary.\nThe general theorem for arbitrary [[closed curves|045YFKRW0P]] without self-intersections is usually proved by approximating the curve by polygons or by homology.\nThe polygonal case is what is used for simple polygons in [[polygon triangulation|Q6KKBP3RRP]] and other parts of computational geometry.\n",
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