Convex and concave polygons

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An example of a convex polygon: a regular pentagon

In geometry, a polygon can be either convex or concave (non-convex or reentrant).

Convex polygons[edit]

A convex polygon is a simple polygon whose interior is a convex set.[1] The following properties of a simple polygon are both equivalent to convexity:

A simple polygon is strictly convex if every internal angle is strictly less than 180 degrees. Equivalently, a polygon is strictly convex if every line segment between two nonadjacent vertices of the polygon is strictly interior to the polygon except at its endpoints.

Every nondegenerate triangle is strictly convex.

Concave or non-convex polygons[edit]

An example of a concave polygon.

A simple polygon that is not convex is called concave,[2] non-convex[3] or reentrant.[4] A concave polygon will always have an interior angle with a measure that is greater than 180 degrees.[5]

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).[6]

See also[edit]


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

External links[edit]