Often the term "polygon" is used in the meaning of "closed polygonal chain".

The same algorithm may also be used for determining whether a closed polygonal chain forms a simple polygon.

A simple closed polygonal chain in the plane is the boundary of a simple polygon.

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

A knot is tame if and only if it can be represented as a finite closed polygonal chain.

Every nontrivial monotone polygonal chain is open.

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.

For closed polygonal chains the same result holds with the integral of curvature replaced by the sum of angles between adjacent segments of the chain.

Mathematicians are often concerned only with the closed polygonal chain and with simple polygons which do not self-intersect, and may define a polygon accordingly.

It turns out that the resulting polygonal chain will in fact be a convex polygon which is the Minkowski sum of P and Q.

A polygonal chain, polygonal curve, polygonal path, or piecewise linear curve, is a connected series of line segments.

Further, piecewise functions such as splines and polygonal chains are common in mathematics, and PDIFF provides a category for discussing them.

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.

More generally, polygonal chains that do not go back on themselves (no 180 angles) have well-defined total curvature, interpreting the curvature as point masses at the angles.

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.

By approximating arbitrary curves by polygonal chains one may extend the definition of total curvature to larger classes of curves, within which the Fary-Milnor theorem also holds (, ).

A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain).

If a closed polygonal chain embedded in the plane divides it into two regions one of which is topologically equivalent to a disk, then the chain is called a weakly simple polygon.

More formally, a polygonal chain P is a curve specified by a sequence of points called its vertices so that the curve consists of the line segments connecting the consecutive vertices.

Although originally defined only for straight-line drawings of graphs, later authors have also investigated the angular resolution of drawings in which the edges are polygonal chains, circular arcs, or spline curves.

In two dimensions, the convex hull is sometimes partitioned into two polygonal chains, the upper hull and the lower hull, stretching between the leftmost and rightmost points of the hull.

The k-level of an arrangement is the polygonal chain formed by the edges that have exactly k other lines directly below them, and the k-level is the portion of the arrangement below the k-level.

For example, referring to the image above, the polygonal chain ABCBA is a weakly simple polygon: it may be viewed as the limit of "squeezing" of the polygon ABCFGHA.

To avoid intersections between vertices and edges, edges that span multiple layers of the drawing may be drawn as polygonal chains or spline curves passing through each of the positions assigned to the dummy vertices along the edge.