Convex and concave polygons

Convex and concave polygons
An example of a convex polygon: a regular pentagon

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

Contents

Convex polygons

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

A simple polygon is strictly convex if every internal angle is strictly less than 180 degrees.[citation needed] 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.[citation needed]

Every nondegenerate triangle is strictly convex.

Concave or non-convex polygons

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.[citation needed]

It is always possible to cut 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]

See also

References

  1. ^ Definition and properties of convex polygons with interactive animation.
  2. ^ McConnell, Jeffrey J. (2006), Computer Graphics: Theory Into Practice, p. 130, ISBN 0763722502 .
  3. ^ Leff, Lawrence (2008). Let's Review: Geometry. Hauppauge, NY: Barron's Educational Series. pp. 66. ISBN 9780764140693. 
  4. ^ Mason, J. I. (1935), "On the angles of a polygon", The Mathematical Gazette (The Mathematical Association) 30 (291): 237–238, doi:10.2307/3611229, JSTOR 3611229 .
  5. ^ Chazelle, Bernard; Dobkin, David P. (1985), "Optimal convex decompositions", in Toussaint, G. T., Computational Geometry, Elsevier, pp. 63–133, http://www.cs.princeton.edu/~chazelle/pubs/OptimalConvexDecomp.pdf .

External links


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