segmentides

Segmentides are a geometric construct used in planning and graph-theoretic representations of networks. In its common form, a segmentide is a finite union of closed line segments in the plane arranged so that any two segments meet only at endpoints, and the overall set is connected. This yields a shape that resembles a network or polyline made of straight, non-overlapping pieces.

The internal structure of a segmentide can be analyzed via its endpoint set and the incidence of segments at those endpoints. Each endpoint acts as a vertex, and each segment acts as an edge in an underlying graph. A point where multiple segments meet becomes a multi-degree vertex, while a single, non-branching segment forms a simple edge. This interpretation makes segmentides closely related to planar straight-line graphs and to polyline networks used in computer graphics and geographic information systems.

Variants of segmentides include open and closed forms, simple segmentides (with no branching), and those containing cycles. In three-dimensional space, the concept generalizes to finite unions of line segments with similar intersection restrictions, producing 1-dimensional skeletal networks in space.

Construction and analysis questions for segmentides include decomposing a given shape into constituent segments, determining total length, counting vertices, and studying properties such as connectivity and planarity. Applications appear in modeling road and utility networks, architectural and microfabrication layouts, and various algorithms in computational geometry and graph drawing.

See also: segment, polyline, planar straight-line graph, segment graph, geometric graph.