A polygonal curve is a simple closed path consisting of a finite sequence of coplanar points together with each open interval determined by two consecutive points on the path. A polygon is a closed planar region bounded by a polygonal curve. Each closed interval in the polygonal curve is called an edge or side of the polygon, and each point in the sequence of points determining the polygonal curve is called a vertex of the polygon.

A polygon with sides is called an -gon, although for small there are more traditional names:

At each vertex, the two sides that meet determine two angles: the interior angle and the exterior angle. The former angle opens towards the interior of the polygon, and the latter towards the exterior of the polygon.

Below are some properties for polygons.

1. In a Euclidean space, the sum of the interior angles of an -gon is .
2. The boundary of a polygon divides the plane into two connected components, one bounded (the interior of the polygon) and one unbounded. This result is the Jordan curve theorem for polygons. Moise proves this directly in [3], pp. 16-16.
3. In complex analysis, the Schwarz-Christoffel transformation [4] gives a conformal map from any polygon to the upper half plane.
4. The area of a lattice polygon can be calculated using Pick's theorem.

Alternative definitions

A concept such as the polygon permits various equivalent definitions. Here are a few alternative definitions.

1. A triangle is the convex hull of three noncollinear points. A union of a finite positive number of coplanar triangles with disjoint interiors is a polygon if its interior is connected.
2. A polygon is a set homeomorphic to the closed unit disc whose boundary is a finite union of line segments.
3. A polygon is a 2-dimensional polytope (not necessarily convex).

Some authors do not include the interior of a polygon as part of the polygon, and thus identify a polygon with what we call a polygonal curve. Such authors sometimes remove the requirement that the path determining a polygon be simple.

Bibliography

1

K. Borsuk and W. Szmielew, Foundations of Geometry, North-Holland Publishing Company, 1960.

2

H.G. Forder, The Foundations of Euclidean Geometry, Dover Publications, 1958.

3

E.E. Moise, Geometric Topology in Dimensions 2 and 3, Springer-Verlag, 1977.

4

R.A. Silverman, Introductory Complex Analysis, Dover Publications, 1972.