# Polygonal chain

A simple polygonal chain
A self-intersecting polygonal chain
A closed polygonal chain

A polygonal chain, polygonal curve, polygonal path, or piecewise linear curve, is a connected series of line segments. More formally, a polygonal chain P is a curve specified by a sequence of points $\scriptstyle(A_1, A_2, \dots, A_n)$ called its vertices so that the curve consists of the line segments connecting the consecutive vertices.

In computer graphics a polygonal chain is called a polyline and is often used to approximate curved paths.

A simple polygonal chain is one in which only consecutive (or the first and the last) segments intersect and only at their endpoints.

A closed polygonal chain is one in which the first vertex coincides with the last one, or, alternatively, the first and the last vertices are also connected by a line segment. A simple closed polygonal chain in the plane is the boundary of a simple polygon. Often the term "polygon" is used in the meaning of "closed polygonal chain".

In some cases it is important to draw a distinction between a polygonal area and a polygonal chain.

A polygonal chain is called monotone, if there is a straight line L such that every line perpendicular to L intersects the chain at most once. Every nontrivial monotone polygonal chain is open. Compare with "monotone polygon". If that straight line is horizontal, then the polygonal chain is a piecewise linear function.

A set of n=17 points has a polygonal path with 4 same-sign slopes

In any set of at least n points, we can find a polygonal path of at least ⌊√(n-1)⌋ edges in which all slopes have the same sign. This is a corollary of the Erdős–Szekeres theorem.

## Application

Polygonal curve approximation: the unknown curve is in blue, and a polygonal approximation is in red.

Polygonal curves can be used to approximate other curves and boundaries of real-life objects.