In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain ⁠${\displaystyle P}$⁠ is a curve specified by a sequence of points ${\displaystyle (A_{1},A_{2},\dots ,A_{n})}$ called its vertices. The curve itself consists of the line segments connecting the consecutive vertices.

A simple polygonal chain is one in which only consecutive 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", but in some cases it is important to draw a distinction between a polygonal area and a polygonal chain. A space closed polygonal chain is also known as a skew "polygon".

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. In comparison, a monotone polygon is a polygon (a closed chain) that can be partitioned into exactly two monotone chains. The graphs of piecewise linear functions form monotone chains with respect to a horizontal line.

Each segment of a polygonal chain is typically parametrized linearly, using linear interpolation between successive vertices. For the whole chain, two parametrizations are common in practical applications: Each segment of the chain can be assigned a unit interval of the parameter corresponding to the index of the first vertex; alternately, each segment of the chain can be assigned an interval of the parameter corresponding to the length of the segment, so that the parameter corresponds uniformly to arclength along the whole chain.

Every set of at least ${\displaystyle n}$ points contains a polygonal path of at least ${\displaystyle \lfloor {\sqrt {n-1}}\rfloor }$ edges in which all slopes have the same sign. This is a corollary of the Erdős–Szekeres theorem.

Polygonal chains can often be used to approximate more complex curves. In this context, the Ramer–Douglas–Peucker algorithm can be used to find a polygonal chain with few segments that serves as an accurate approximation.

In graph drawing, polygonal chains are often used to represent the edges of graphs, in drawing styles where drawing the edges as straight line segments would cause crossings, edge-vertex collisions, or other undesired features. In this context, it is often desired to draw edges with as few segments and bends as possible, to reduce the visual clutter in the drawing; the problem of minimizing the number of bends is called bend minimization.