Joonsegment

Joonsegment is a concept in metric geometry and related fields that generalizes the idea of a line segment between two points in a space equipped with a distance function. In a metric space (X,d), the joonsegment J(a,b) between points a and b is defined as the union of all points that lie on at least one length-minimizing path from a to b. Equivalently, it is the set of points that belong to some shortest path joining a to b. In Euclidean space, where the shortest paths are straight lines, J(a,b) coincides with the ordinary line segment [a,b]. In spaces that admit multiple geodesics between a and b, J(a,b) can be larger, consisting of the points on all such geodesics.

History and terminology clarify that the term was introduced in the mathematical literature to study generalized convexity and path-connectedness in non-Euclidean spaces. The concept has since appeared in work on geodesic convexity, network routing, and geometric analysis, where it serves as a natural extension of the notion of a segment to more complex spaces.

Computation and examples vary by setting. In graphs, J(a,b) is the subgraph formed by all vertices and edges that lie on at least one shortest path between a and b; in weighted graphs, algorithms based on shortest-path trees or Yen’s algorithm can be used to enumerate or approximate the joonsegment. In manifolds and other continuous spaces, visualization and computation rely on characterizing all geodesics between the endpoints.

Applications include robotics path planning, computer graphics, geographic information systems, and the study of generalized convexity in non-Euclidean spaces.

See also: geodesic, line segment, shortest path, convex hull, geodesic convexity.