Simple polygon

In geometry, a simple polygon is a polygon whose sides do not intersect. They are also called Jordan polygons, because the Jordan curve theorem can be used to prove that such a polygon divides the plane into two regions, the region inside it and the region outside it. A simple polygon is topologically equivalent to a disk.

A polygon that is not simple is called "self-intersecting" by geometers and "complex" by computer graphics programmers (in geometry, a complex polygon is something different). Such a polygon does not necessarily have a well-defined inside and outside.

In computational geometry, there are several important problems where the given input is a simple polygon, each depending critically on its well-defined interior:
* Determining if a point falls inside a simple polygon; see Point in polygon
* Finding the area contained by a simple polygon; see Polygon area
* Polygon triangulation: dividing a simple polygon into triangles. Although convex polygons are easy to triangulate, triangulating a general simple polygon is more difficult because we have to avoid adding edges that cross outside the polygon. Nevertheless, Bernard Chazelle showed in 1991 that any simple polygon with "n" vertices can be triangulated in Θ("n") time, which is optimal.
* Polygon union: finding the simple polygon or polygons containing the area inside either of two simple polygons
* Polygon intersection: finding the simple polygon or polygons containing the area inside both of two simple polygons
*Convex hull of a simple polygon.

External links

*Mathworld | urlname=SimplePolygon | title=Simple polygon