segmentilles

Segmentilles are a hypothetical geometric construct consisting of a finite chain of straight line segments connected end-to-end in the plane. The term blends segment with the suffix -ille, signaling a small or simple assembly of segments. In this framework, a segmentille of order n is the union of n consecutive unit segments joined at their endpoints; a segmentille can be open (a polyline with two endpoints) or closed (a polygonal cycle). They are primarily used as a discrete model for approximating curves or paths.

Formal notation and variants: A segmentille S(n, {θ1,...,θn}) can be defined by the directions of its consecutive segments, where θi denotes the orientation of the i-th unit segment. Variants include oriented segmentilles with weights, and smooth-approximation segmentilles that use increasingly short segments to better fit a given curve.

Properties: Segmentilles are piecewise-linear, finite in length, and have a turning angle equal to the sum of the exterior angles at the joints. They can exactly represent straight lines when all directions are equal, and approximate smooth curves as n grows.

Origins and usage: The term was introduced in theoretical discussions of discrete geometry and computer graphics to study polygonal representations of curves and paths. Although not standard terminology in formal geometry, segmentilles appear in expository material and algorithmic contexts.

Applications: Used for stroke rendering in vector graphics, polygonal modeling, and path discretization in robotics and GIS workflows. They provide a simple, uniform representation for chaining linear elements.

See also: polyline, polygonal chain, digital geometry, polyline simplification.