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).

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:

• Every internal angle is less than or equal to 180 degrees.^{[citation needed]}
• Every line segment between two vertices remains inside or on the boundary of the polygon.^{[citation needed]}

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

• Convex hull

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

• Weisstein, Eric W., "Convex polygon" from MathWorld.
• Weisstein, Eric W., "Concave polygon" from MathWorld.
• http://www.rustycode.com/tutorials/convex.html