segmentssubsets

Segmentssubsets is a concept in discrete and computational geometry that studies the subsets formed from a finite collection of line segments in the plane. It focuses on how unions of portions of these segments can be combined, decomposed, and analyzed from a combinatorial and geometric perspective. The term is used to describe the family of subsets that can be constructed from the original segments by taking finite unions of their subsegments, and it often intersects with questions about coverage, visibility, and the complexity of geometrical arrangements.

Formally, let F = {s1, s2, ..., sn} be a finite set of closed line segments in the Euclidean plane. A segment-subset is any subset of the plane that can be written as a finite union of closed subsegments of members of F. A subsegment is a connected portion of a segment. The collection of all segment-subsets, usually denoted L(F), is closed under union and intersection, and is ordered by set inclusion. This yields a finite lattice or algebra of subsets generated by the segments in F, with the union corresponding to the join operation and intersection to the meet.

Examples include the full set of a crossing pair of segments, a single arm of a cross, or any combination of subsegments drawn from the original segments. The study often emphasizes the boundary structure, the number of connected components, and the total length of unions, as well as the complexity that arises from overlaps and interactions among segments.

Applications of segmentssubsets appear in coverage and sensing problems, digital geometry, and computer graphics, where efficient representation and manipulation of unions of line primitives are relevant. Related areas include segment intersection graphs, line arrangements, and the broader study of subset lattices generated by geometric objects.