Simple polygon

That is, it is a piecewise-linear Jordan curve consisting of finitely many line segments.

The sum of external angles of a simple polygon is

of its diagonals, and by the art gallery theorem its interior is visible from some

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.

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.

[3] The line segments that form a polygon are called its edges or sides.

An endpoint of a segment is called a vertex (plural: vertices)[2] 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.

[2] Some sources allow two line segments to form a straight angle (180°),[5] while others disallow this, instead requiring collinear segments of a closed polygonal chain to be merged into a single longer side.

[8] Indeed, Camille Jordan's original proof of this theorem took the special case of simple polygons (stated without proof) as its starting point.

[9] The region inside the polygon (its interior) forms a bounded set[2] topologically equivalent to an open disk by the Jordan–Schönflies theorem,[10] with a finite but nonzero area.

[11] The polygon itself is topologically equivalent to a circle,[12] and the region outside (the exterior) is an unbounded connected open set, with infinite area.

[11] Although the formal definition of a simple polygon is typically as a system of line segments, it is also possible (and common in informal usage) to define a simple polygon as a closed set in the plane, the union of these line segments with the interior of the polygon.

For every simple polygon, the sum of the external angles is

Thus the sum of the internal angles, for a simple polygon with

[14] Every simple polygon can be partitioned into non-overlapping triangles by a subset of its diagonals.

[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.

[16] According to the art gallery theorem, in a simple polygon with

to a selected vertex, passing only through interior points of the polygon.

[2] A monotone polygon, with respect to a straight line

intersects the interior of the polygon in a connected set.

Other computational problems studied for simple polygons include constructions of the longest diagonal or the longest line segment interior to a polygon,[13] of the convex skull (the largest convex polygon within the given simple polygon),[29][30] and of various one-dimensional skeletons approximating its shape, including the medial axis[31] and straight skeleton.

They can be defined in a way that always produces a two-dimensional region, but this requires careful definitions of the intersection and difference operations in order to avoid creating one-dimensional features or isolated points.

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.

[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 traveling salesperson problem.

[37] Every simple polygon can be represented by a formula in constructive solid geometry that constructs the polygon (as a closed set including the interior) from unions and intersections of half-planes, with each side of the polygon appearing once as a half-plane in the formula.

The computational complexity of reconstructing a polygon that has a given graph as its visibility graph, with a specified Hamiltonian cycle as its cycle of sides, remains an open problem.

Two simple polygons (green and blue) and a self-intersecting polygon (red, in the lower right, not simple)
Parts of a simple polygon
A triangulated polygon with 11 vertices: 11 sides and 8 diagonals form 9 triangles.
This 42-vertex polygonal art gallery is entirely visible from cameras placed at the 4 marked vertices.
To test whether a point is inside the polygon, construct any ray emanating from the point and count its intersections with the polygon. If it crosses only interior points of edges, an odd number of times, the point lies inside the polygon; if even, the point lies outside. Rays through polygon vertices or containing its edges need special care. [ 19 ]
A simple polygon (interior shaded blue) and its convex hull (surrounding both blue and yellow regions)
The black polygon is the shortest loop connecting every red dot, a solution to the traveling salesperson problem.