Simple polygon

${\displaystyle \pi -\theta }$

${\displaystyle 2\pi }$

${\displaystyle n}$

${\displaystyle (n-2)\pi }$

Every simple polygon can be partitioned into non-overlapping triangles by a subset of its diagonals. When the polygon has ${\displaystyle n}$ sides, this produces ${\displaystyle n-2}$ triangles, separated by ${\displaystyle n-3}$ diagonals. The resulting partition is called a polygon triangulation.^[8] The shape of a triangulated simple polygon can be uniquely determined by the internal angles of the polygon and by the cross-ratios of the quadrilaterals formed by pairs of triangles that share a diagonal.^[15]

According to the two ears theorem, every simple polygon that is not a triangle has at least two ears, vertices whose two neighbors are the endpoints of a diagonal.^[8] A related theorem states that every simple polygon that is not a convex polygon has a mouth, a vertex whose two neighbors are the endpoints of a line segment that is otherwise entirely exterior to the polygon. The polygons that have exactly two ears and one mouth are called anthropomorphic polygons.^[16]

According to the

${\displaystyle n}$

${\displaystyle \lfloor n/3\rfloor }$

${\displaystyle p}$

${\displaystyle p}$

${\displaystyle \lfloor n/3\rfloor }$

Every convex polygon is a simple polygon. Another important class of simple polygons are the star-shaped polygons, the polygons that have a point (interior or on their boundary) from which every point is visible.^[2]

A monotone polygon, with respect to a straight line ${\displaystyle L}$, is a polygon for which every line perpendicular to ${\displaystyle L}$ intersects the interior of the polygon in a connected set. Equivalently, it is a polygon whose boundary can be partitioned into two monotone polygonal chains, subsequences of edges whose vertices, when projected perpendicularly onto ${\displaystyle L}$, have the same order along ${\displaystyle L}$ as they do in the chain.^[18]

In computational geometry, several important computational tasks involve inputs in the form of a simple polygon.

• Point in polygon testing involves determining, for a simple polygon ${\displaystyle P}$ and a query point ${\displaystyle q}$, whether ${\displaystyle q}$ lies interior to ${\displaystyle P}$. It can be solved in
  linear time; alternatively, it is possible to process a given polygon into a data structure, in linear time, so that subsequent point in polygon tests can be performed in logarithmic time.^[20]
• Simple formulae are known for computing the area of the interior of a polygon. These include the shoelace formula for arbitrary polygons,^[21] and Pick's theorem for polygons with integer vertex coordinates.^[12][22]
• The convex hull of a simple polygon can also be found in linear time, faster than algorithms for finding convex hulls of points that have not been connected into a polygon.^[6]
• Constructing a triangulation of a simple polygon can also be performed in linear time, although the algorithm is complicated. A modification of the same algorithm can also be used to test whether a closed polygonal chain forms a simple polygon (that is, whether it avoids self-intersections) in linear time.^[23] This also leads to a linear time algorithm for solving the art gallery problem using at most ${\displaystyle \lfloor n/3\rfloor }$ points, although not necessarily using the optimal number of points for a given polygon.
  NP-complete.^[25]
• A geodesic path,^[26] the shortest path in the plane that connects two points interior to a polygon, without crossing to the exterior, may be found in linear time by an algorithm that uses triangulation as a subroutine.^[27] The same is true for the geodesic center, a point in the polygon that minimizes the maximum length of its geodesic paths to all other points.^[26]
• The visibility polygon of an interior point of a simple polygon, the points that are directly visible from the given point by line segments interior to the polygon, can be constructed in linear time.^[28] The same is true for the subset that is visible from at least one point of a given line segment.^[27]

Other computational problems studied for simple polygons include constructions of the longest diagonal or the longest line segment interior to a polygon,

According to the Riemann mapping theorem, any simply connected open subset of the plane can be conformally mapped onto a disk. Schwarz–Christoffel mapping provides a method to explicitly construct a map from a disk to any simple polygon using specified vertex angles and pre-images of the polygon vertices on the boundary of the disk. These pre-vertices are typically computed numerically.^[35]

Every finite set of points in the plane that does not lie on a single line can be connected to form the vertices of a simple polygon (allowing 180° angles); for instance, one such polygon is the solution to the

Every simple polygon can be represented by a formula in

half-planes

, with each side of the polygon appearing once as a half-plane in the formula. Converting an ${\displaystyle n}$-sided polygon into this representation can be performed in time ${\displaystyle O(n\log n)}$.

The

• Carpenter's rule problem, on continuous motion of a simple polygon into a convex polygon
• Erdős–Nagy theorem, a process of reflecting pockets of a non-convex simple polygon to make it convex
• Net (polyhedron), a simple polygon that can be folded and glued to form a given polyhedron
• Spherical polygon
  , an analogous concept on the surface of a sphere
• Weakly simple polygon, a generalization of simple polygons allowing the edges to touch in limited ways