Concave polygon

An example of a concave polygon.

A simple polygon that is not convex is called concave,[1] non-convex[2] or reentrant.[3] A simple concave polygon will always have an interior angle with a measure that is greater than 180 degrees.[4] Concave can be remembered by students using one of two easy tricks. First, students can say that one side looks to have "caved" in. The other way is to think of a cave in a mountain side. If you see an entrance to a cave, then the polygon must be concave.

It is always possible to partition a concave polygon into a set of convex polygons. A polynomial-time algorithm for finding a decomposition into as few convex polygons as possible is described by Chazelle & Dobkin (1985).[5]

Notes

  1. ↑ McConnell, Jeffrey J. (2006), Computer Graphics: Theory Into Practice, p. 130, ISBN 0-7637-2250-2.
  2. ↑ Leff, Lawrence (2008), Let's Review: Geometry, Hauppauge, NY: Barron's Educational Series, p. 66, ISBN 978-0-7641-4069-3
  3. ↑ Mason, J.I. (1946), "On the angles of a polygon", The Mathematical Gazette (The Mathematical Association) 30 (291): 237–238, JSTOR 3611229.
  4. ↑ Definition and properties of concave polygons with interactive animation.
  5. ↑ Chazelle, Bernard; Dobkin, David P. (1985), "Optimal convex decompositions", in Toussaint, G.T., Computational Geometry (PDF), Elsevier, pp. 63–133.

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