Linearwege

Linearwege is a term used in mathematics and computer science to denote a class of paths that exhibit linear or piecewise-linear structure. The word is derived from German, where weg means “path” or “way” and linear indicates a linear form or direction.

In geometry, a Linearweg refers to a path in Euclidean space that is piecewise linear: it consists of a finite sequence of straight-line segments joined at vertices. The path is straight within each segment, and the overall trajectory may change direction only at the segment junctions. A degenerate Linearweg is a single straight edge.

In graph theory and routing, Linearwege can have multiple related meanings. One common interpretation is a path whose traversal respects a given linear order of the vertices, yielding a monotone path with respect to that order. Another usage describes a path that minimizes a linear cost function defined on the edges, i.e., a shortest path under linear weights. In both senses, Linearwege are often studied for their simplicity and tractability compared to general curved or highly irregular routes.

Computationally, Linearwege are well-suited to representation as polylines or as sequences of vertices. They can be computed using standard shortest-path algorithms with appropriate edge weights or by constructing piecewise-linear approximations in motion-planning problems.

Examples include a two-segment path in the plane from (0,0) to (3,4) via an intermediate vertex, or a monotone grid path that never decreases in a chosen coordinate. Applications span geometric modeling, computer graphics, robotics motion planning, and network routing.

See also: piecewise-linear path, shortest path, monotone path, geodesic, polyline.