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Simple polygon
 
 
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In geometryGeometry

Geometry arose as the field of knowledge dealing with spatial relationships....
, a simple polygon is a polygon whose sides do not intersect. They are also called Jordan polygons, because the Jordan curve theoremFacts About Jordan curve theorem

In topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an "i...
 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 diskDisk (mathematics)

In geometry, a disk is the region in a plane contained by a circle....
.

A polygon that is not simple is called a complex polygonComplex polygon

A complex polygon is a polygon which intersects itself....
, such a polygon does not always have a well-defined inside and outside.

In computational geometryComputational geometry

In computer science, computational geometry is the study of algorithms to solve problems stated in terms of 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 polygonPoint in polygon Summary

    In computational geometry, the point-in-polygon problem asks whether a given point in the plane lies inside, outside, or on ...
  • Finding the area contained by a simple polygon; see Polygon area
  • Polygon triangulationPolygon triangulation

    In computational geometry, polygon triangulation is the decomposition of a polygon into a set of triangles....
    : 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 ChazelleBernard Chazelle

    techniques based on [[discrepancy theory]...
     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 hullConvex hull

    In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal c...
     of a simple polygon.

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