Polylinean

Polylinean (plural polylineans) is a term used in geometry and related fields to describe a polygonal chain, i.e., a piecewise-linear curve formed by a finite sequence of straight segments connected at vertices. A polylinean can be open or closed; when closed and non-self-intersecting, it is commonly called a polygon.

Formal definition: A polylinean is determined by an ordered list of vertices V = (v0, v1, ..., vn) in a Euclidean space, with edges ei = (vi, vi+1) for i = 0, ..., n-1. The geometric realization is the union of the segments [vi, vi+1]. If v0 = vn, the polylinean is closed; if no two nonconsecutive edges intersect, it is simple.

Properties and variants: The length is L = sum ||vi+1 - vi||. The orientation follows the vertex order. In 3D, polylineans lie in R^3; in 2D they lie in the plane. A polylinean is simple if its edges do not cross; a polygon is a simple, closed polylinean.

Applications: Polylineans model digital curves in computer graphics, geographic information systems (GIS), computer-aided design (CAD), and robotics. They approximate smooth shapes, encode trajectories, and serve as inputs for rendering, vectorization, and path planning. Algorithms exist to simplify, resample, or smooth polylineans, such as polygonal simplification and polyline resampling.

Etymology and usage: The term polylinean is a neologism formed from polyline with the suffix -an to indicate relation or belonging. It is not widely standardized; in many contexts the clearer terms polygonal chain or polyline are preferred. Some discussions use polylinean to emphasize the study or class of polylines.

See also: Polyline, Polygon, Polygonal chain, Piecewise-linear.