Editing Simple polygon (section)

==Definitions==
[[File:Parts of a simple polygon.png|thumb|upright=1.25|Parts of a simple polygon]]
A simple polygon is a [[closed curve]] in the [[Euclidean plane]] consisting of [[line segment|straight line segments]], meeting end-to-end to form a [[polygonal chain]].{{r|milnor}} Two line segments meet at every endpoint, and there are no other points of intersection between the line segments. No [[proper subset]] of the line segments has the same properties.{{r|preparata-shamos}} The qualifier ''simple'' is sometimes omitted, with the word ''polygon'' assumed to mean a simple polygon.{{r|everett-corneil}}

The line segments that form a polygon are called its ''edges'' or ''sides''. An endpoint of a segment is called a ''[[Vertex (geometry)|vertex]]'' (plural: vertices){{r|preparata-shamos}} or a ''corner''. ''Edges'' and ''vertices'' are more formal, but may be ambiguous in contexts that also involve the edges and vertices of a [[Graph (graph theory)|graph]]; the more colloquial terms ''sides'' and ''corners'' can be used to avoid this ambiguity.{{r|compatible}} The number of edges always equals the number of vertices.{{r|preparata-shamos}} Some sources allow two line segments to form a [[straight angle]] (180°),{{r|malkevitch}} while others disallow this, instead requiring collinear segments of a closed polygonal chain to be merged into a single longer side.{{r|mccallum-avis}} Two vertices are ''neighbors'' if they are the two endpoints of one of the sides of the polygon.{{r|4marks}}

Simple polygons are sometimes called '''Jordan polygons''', because they are [[Jordan curve]]s; the [[Jordan curve theorem]] can be used to prove that such a polygon divides the plane into two regions.{{r|meisters}} Indeed, [[Camille Jordan]]'s original proof of this theorem took the special case of simple polygons (stated without proof) as its starting point.{{r|hales}} The region inside the polygon (its ''interior'') forms a [[bounded set]]{{r|preparata-shamos}} [[Homeomorphism|topologically equivalent]] to an [[open disk]] by the [[Jordan–Schönflies theorem]],{{r|thomassen}} with a finite but nonzero [[area]].{{r|margalit-knott}} The polygon itself is topologically equivalent to a [[circle]],{{r|niven-zuckerman}} and the region outside (the ''exterior'') is an unbounded [[Domain (mathematical analysis)|connected open set]], with infinite area.{{r|margalit-knott}} Although the formal definition of a simple polygon is typically as a system of line segments, it is also possible (and common in informal usage) to define a simple polygon as a [[closed set]] in the plane, the union of these line segments with the interior of the polygon.{{r|preparata-shamos}}

A ''diagonal'' of a simple polygon is any line segment that has two polygon vertices as its endpoints, and that otherwise is entirely interior to the polygon.{{r|aggarwal-suri}}