Math 2.0

Geometry Theorem

Jordan curve theorem for polygons

Builds on Closed curveConnected space

Statement

Let C⊂ℝ2C \subset \R^2 be a simple closed polygonal curve. That is, CC is a closed polygonal chain that meets itself only where its last side ends at the start of its first side. Then ℝ2∖C\R^2 \setminus C has exactly two connected components. One component is bounded and the other is unbounded, and CC is the boundary of each of them.

Remarks

The proof uses parity of crossings. For a point x∉Cx \notin C, take a ray from xx in a direction that avoids the vertices of CC, and count how many times it crosses CC. The parity of this count does not depend on the ray, and it is constant along paths that avoid CC. So points of even and odd parity lie in different components. Conversely, two points of the same parity can be joined by a path that runs close to CC, so there are exactly two components. Points far away have even parity and form the unbounded one.

This crossing count is also the standard test of whether a point lies inside a polygon.

Going deeper

For a student

This is the special case of the Jordan curve theorem for polygonal curves, and its proof is elementary. The general theorem for arbitrary closed curves without self-intersections is usually proved by approximating the curve by polygons or by homology. The polygonal case is what is used for simple polygons in polygon triangulation and other parts of computational geometry.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Connected space, concept, no evidence levelConnected spaceconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptVector-valued function, concept, no evidence levelVector-valued functionconceptParametrised curve, concept, no evidence levelParametrised curveconceptClosed curve, concept, no evidence levelClosed curveconceptJordan curve theorem for polygons, statement, no evidence levelJordan curve theorem …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Connected space concept
  2. Euclidean space object
  3. Function concept
  4. Vector-valued function concept
  5. Parametrised curve concept
  6. Closed curve concept
  7. Jordan curve theorem for polygons statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Jordan_curve_theorem
About this page
TagQS9PG4MNTR
URIm2://QS9PG4MNTR
Kindstatement, theorem
WritingW0 stub
MSC51M04, 54D05
StatusThis copy of the site has no build data. Run m2 status QS9PG4MNTR.
AboutClosed curve
UsesClosed curve, Connected space

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.