Weglängen

Weglängen is a German term commonly translated as "path lengths" in English. It is used in mathematics, physics, and related fields to denote the lengths of paths traced within a space or network. In mathematical contexts, the Weglänge of a smooth curve γ is defined as the integral of the speed along the path, typically written as L = ∫ |γ′(t)| dt, for a differentiable γ. In discrete settings such as graphs, the Weglänge of a path is the sum of the edge lengths or weights along that path.

The concept appears in geometry and calculus as a measure of how long a trajectory is between two points, independent of the specific route taken. In navigation and geography, Weglängen describe the length of routes in road or transit networks, contributing to route comparison and distance calculations. In physics and engineering, a related idea is the optical path length, which combines geometric length with a refractive index along the path and is relevant to principles of wave propagation and Fermat’s principle.

Usage notes emphasize the plural form Weglängen when referring to multiple path lengths, and the singular Weglänge for a single path length. The term is often clarified by specifying the context—geometric length, optical path length, or network distance—to avoid ambiguity. The concept is closely linked to distance, metric, and shortest-path problems in mathematics, computer science, and transportation planning.