Simple polygon

In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a simple polygon is a <a href="/facts/Polygon/xvkmYhvG">polygon</a> that does not <a href="/facts/Intersection_(Euclidean_geometry)/RwH57yPq">intersect</a> itself and has no holes. That is, it is a <a href="/facts/Piecewise_linear_curve/GRQ6BMQS">piecewise-linear</a> <a href="/facts/Jordan_curve/zKpty4QT">Jordan curve</a> consisting of finitely many <a href="/facts/Line_segment/V8E0M4qk">line segments</a>. These polygons include as special cases the <a href="/facts/Convex_polygon/7x35w610">convex polygons</a>, <a href="/facts/Star-shaped_polygon/yW0cSRzz">star-shaped polygons</a>, and <a href="/facts/Monotone_polygon/16N42zJL">monotone polygons</a>.
The sum of <a href="/facts/External_angle/jd78Qkbt">external angles</a> of a simple polygon is 
 
 
 
 2
 π
 
 
 {\displaystyle 2\pi }
 
. Every simple polygon with 
 
 
 
 n
 
 
 {\displaystyle n}
 
 sides can be <a href="/facts/Polygon_triangulation/FF0205ye">triangulated</a> by 
 
 
 
 n
 −
 3
 
 
 {\displaystyle n-3}
 
 of its diagonals, and by the <a href="/facts/Art_gallery_theorem/Ytg00bIU">art gallery theorem</a> its interior is visible from some 
 
 
 
 ⌊
 n
 
 /
 
 3
 ⌋
 
 
 {\displaystyle \lfloor n/3\rfloor }
 
 of its vertices.
Simple polygons are commonly seen as the input to <a href="/facts/Computational_geometry/eeaotQtl">computational geometry</a> problems, including <a href="/facts/Point_in_polygon/acr2LQ1n">point in polygon</a> testing, <a href="/facts/Area/KsNG1G9t">area</a> computation, the <a href="/facts/Convex_hull_of_a_simple_polygon/ssfd5A1g">convex hull of a simple polygon</a>, triangulation, and <a href="/facts/Euclidean_shortest_path/eAKJgenE">Euclidean shortest paths</a>.
Other constructions in geometry related to simple polygons include <a href="/facts/Schwarz%25E2%2580%2593Christoffel_mapping/vzHbRDDG">Schwarz–Christoffel mapping</a>, used to find <a href="/facts/Conformal_map/cEpQnMad">conformal maps</a> involving simple polygons, <a href="/facts/Polygonalization/4YArjFyH">polygonalization</a> of point sets, <a href="/facts/Constructive_solid_geometry/Dcs2fzyU">constructive solid geometry</a> formulas for polygons, and <a href="/facts/Visibility_graph/kEaHTj2A">visibility graphs</a> of polygons.

Simple polygon open-in-new

Simple polygon