Simple polygon

In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, they are piecewise-linear Jordan curves consisting of finitely many line segments. They include as special cases the convex polygons, star-shaped polygons, and monotone polygons. The sum of external angles of a simple polygon is . Every simple polygon with sides can be triangulated by of its diagonals, and by the art gallery theorem its interior is visible from some 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. Other than the shared endpoints of consecutive line segments in this chain, no two of the line segments may intersect each other. 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. Exactly two edges meet at each vertex, and 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.