Editing Concave polygon

{{Short description|Simple polygon which is not convex}}
[[File:Simple polygon.svg|thumb|200px|An example of a concave polygon.]]
A [[simple polygon]] that is not [[convex polygon|convex]] is called '''concave''',<ref>{{citation |first=Jeffrey J. |last=McConnell |year=2006 |title=Computer Graphics: Theory Into Practice |isbn=0-7637-2250-2 |page=[https://archive.org/details/computergraphics0000mcco/page/130 130] |url=https://archive.org/details/computergraphics0000mcco/page/130 }}.</ref> '''non-convex'''<ref>{{Citation |last=Leff |first=Lawrence |title=Let's Review: Geometry |year=2008 |publisher=Barron's Educational Series |location=Hauppauge, NY |isbn=978-0-7641-4069-3 |pages=66}}</ref> or '''reentrant'''.<ref>{{citation |first=J.I. |last=Mason |year=1946 |title=On the angles of a polygon |journal=The Mathematical Gazette |volume=30 |issue=291 |jstor=3611229 |pages=237–238 <!--|do=10.2307/3611229--> |publisher=The Mathematical Association|doi=10.2307/3611229 }}.</ref> A concave polygon will always have at least one [[reflex angle|reflex interior angle]]—that is, an angle with a measure that is between 180° degrees and 360° degrees exclusive.<ref name=MOR>{{Cite web | url = http://www.mathopenref.com/polygonconcave.html | title = Definition and properties of concave polygons with interactive animation.}}</ref>

==Polygon==
Some lines containing interior points of a concave polygon intersect its boundary at more than two points.<ref name=MOR/> Some [[diagonal#Polygons|diagonals]] of a concave polygon lie partly or wholly outside the polygon.<ref name=MOR/> Some [[extended side|sidelines]] of a concave polygon fail to divide the plane into two half-planes one of which entirely contains the polygon. None of these three statements holds for a convex polygon.

As with any simple polygon, the sum of the [[internal angle]]s of a concave polygon is {{pi}}(''n''&nbsp;−&nbsp;2) [[radian]]s, equivalently 180°(''n''&nbsp;−&nbsp;2) degrees, where ''n'' is the number of sides.

It is always possible to [[Partition of a set|partition]] a concave polygon into a set of convex polygons. A [[polynomial-time|polynomial-time algorithm]] for finding a decomposition into as few convex polygons as possible is described by {{harvtxt|Chazelle|Dobkin|1985}}.<ref>{{citation |first1=Bernard |last1=Chazelle |author1-link=Bernard Chazelle |first2=David P. |last2=Dobkin |author2-link=David P. Dobkin |contribution=Optimal convex decompositions |title=Computational Geometry |year=1985 |editor-first=G.T. |editor-last=Toussaint |publisher=Elsevier |pages=63–133 |url=http://www.cs.princeton.edu/~chazelle/pubs/OptimalConvexDecomp.pdf}}.</ref>

According to Euclidean geometry, a [[triangle]] can never be concave, but there exist concave polygons with ''n'' sides for any ''n'' > 3. An example of a concave [[quadrilateral]] is the [[dart (geometry)|dart]].

At least one interior angle does not contain all other vertices in its edges and interior.

The [[convex hull]] of the concave polygon's vertices, and that of its edges, contains points that are exterior to the polygon.

==Notes==
{{reflist}}

{{polygons}}
==External links==
*{{mathworld |urlname=ConcavePolygon |title=Concave polygon}}

[[Category:Types of polygons]]