Convex and concave polygons

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In geometry, a polygon can be either convex or concave.

[edit] 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 180 degrees.
• Every line segment between two vertices remains inside or on the boundary of the polygon.

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.

[edit] Concave polygons

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

It is 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).^[4]

[edit] 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. ^ Mason, J. I. (1935), "On the angles of a polygon", The Mathematical Gazette 30 (291): 237–238, http://www.jstor.org/stable/3611229 .
4. ^ 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 .