Polygon

A polygon (from the Greek poly, for "many", and gwnos, for "angle") is a closed planar path composed of a finite number of sequential straight line segments. The term polygon sometimes also describes the interior of the polygon (the open area that this path encloses) or the union of both. The straight line segments that make up the polygon are called its sides or edges and the points where the sides meet are the polygon's vertices.

Names and types

Polygons are named according to the number of sides, combining a Greek root with the suffix -gon, e.g. pentagon, dodecagon. The triangle and quadrilateral are exceptions. For larger numbers, mathematicians write the numeral itself, eg 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula.

Taxonomic classification

The taxonomic classification of polygons is illustrated by the following tree:

                                       Polygon
                                      /       \
                                  Simple     Complex
                                 /     \
                            Convex     Concave
                             /
                        Cyclic 
                        /    
                   Regular

• A polygon is called simple if it is described by a single, non-intersecting boundary; otherwise it is called complex.
• A simple polygon is called convex if it has no internal angles greater than 180° otherwise it is called concave.
• A convex polygon is called concyclic or cyclic polygon if all the vertices lie on a single circle.
• A cyclic polygon is called regular if all its sides are of equal length and all its angles are equal.

Properties

We will assume Euclidean geometry throughout.

Any polygon, regular or irregular, complex or simple, has as many angles as it has sides. The sum of the inner angles of a simple n-gon is (n-2)π radians (or (n-2)180°), and the inner angle of a regular n-gon is (n-2)π/n radians (or (n-2)180°/n). This can be seen in two different ways:

• Moving around a simple n-gon (like a car on a road), the amount one "turns" at a vertex is 180° - the inner angle. "Driving around" the polygon, one makes one full turn, so the sum of these turns must be 360°, from which the formula follows easily. The reasoning also applies if some inner angles are more than 180°: going clockwise around, it means that one sometime turns left instead of right, which is counted as a negative amount one turns.
• Any simple n-gon can be considered to be made up of (n-2) triangles, each of which has an angle sum of π radians or 180°.

The area A of a simple polygon can be computed if the cartesian coordinates (x₁, y₁), (x₂, y₂), ..., (x_n, y_n) of its vertices, listed in order as the area is circulated in counter-clockwise fashion, are known. The formula is

A = 1/2 · (x₁y₂ - x₂y₁ + x₂y₃ - x₃y₂ + ... + x_ny₁ - x₁y_n)

= 1/2 · (x₁(y₂ - y_n) + x₂(y₃ - y₁) + x₃(y₄ - y₂) + ... + x_n(y₁ - y_n-1))

The formula was described by Meister in 1769 and by Gauss in 1795. It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of Green's theorem.

If the polygon can be drawn on an equally-spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points.

If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon. This is the Bolyai-Gerwien theorem.

All regular polygons are concyclic, as are all triangles and rectangles (see circumcircle).

A regular n-sided polygon can be constructed with ruler and compass if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon.

Point in polygon test

In computer graphics and computational geometry, it is often necessary to determine whether a given point P = (x₀,y₀) lies inside a simple polygon given by a sequence of line segments. It is known as Point in polygon test.

See also: geometric shape, polyhedron, polytope, cyclic polygon, synthetic geometry.

External link

Optimized Point-in-Polygon code http://www.ecse.rpi.edu/Homepages/wrf/research/geom/pnpoly.html