Simple polygon

In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a piecewise-linear Jordan curve consisting of finitely many line segments. These polygons include as special cases the convex polygons, star-shaped polygons, and monotone polygons.

The sum of external angles of a simple polygon is ${\displaystyle 2\pi }$. Every simple polygon with ${\displaystyle n}$ sides can be triangulated by ${\displaystyle n-3}$ of its diagonals, and by the art gallery theorem its interior is visible from some ${\displaystyle \lfloor n/3\rfloor }$ of its vertices.

Simple polygons are commonly seen as the input to computational geometry problems, including point in polygon testing, area computation, the convex hull of a simple polygon, triangulation, and Euclidean shortest paths.

Other constructions in geometry related to simple polygons include Schwarz–Christoffel mapping, used to find conformal maps involving simple polygons, polygonalization of point sets, constructive solid geometry formulas for polygons, and visibility graphs of polygons.

A simple polygon is a closed curve in the Euclidean plane consisting of straight line segments, meeting end-to-end to form a polygonal chain. 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. The qualifier simple is sometimes omitted, with the word polygon assumed to mean a simple polygon.

The line segments that form a polygon are called its edges or sides. An endpoint of a segment is called a vertex (plural: vertices) 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; the more colloquial terms sides and corners can be used to avoid this ambiguity. The number of edges always equals the number of vertices. Some sources allow two line segments to form a straight angle (180°), while others disallow this, instead requiring collinear segments of a closed polygonal chain to be merged into a single longer side. Two vertices are neighbors if they are the two endpoints of one of the sides of the polygon.

Simple polygons are sometimes called Jordan polygons, because they are Jordan curves; the Jordan curve theorem can be used to prove that such a polygon divides the plane into two regions. Indeed, Camille Jordan's original proof of this theorem took the special case of simple polygons (stated without proof) as its starting point. The region inside the polygon (its interior) forms a bounded set topologically equivalent to an open disk by the Jordan–Schönflies theorem, with a finite but nonzero area. The polygon itself is topologically equivalent to a circle, and the region outside (the exterior) is an unbounded connected open set, with infinite area. 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.

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.