Geodesische

Geodesische (geodesics) are curves that generalize the notion of straight lines to curved spaces. In differential geometry, a geodesic is a curve whose tangent vector is parallel transported along itself, making the path the locally distance-minimizing route between nearby points. Equivalently, it is a stationary point of the length functional; on Riemannian manifolds it satisfies the geodesic equation, which encodes how the space’s curvature directs motion.

Examples: In Euclidean 3-space or on a plane, geodesics are straight lines. On the surface of a sphere, geodesics are great circles like the equator. On a cylinder, geodesics map to straight lines on the unrolled rectangle, appearing as helices when wrapped around the surface. In more general spaces, geodesics depend on the metric.

Properties: Geodesics are locally distance-minimizing, but not necessarily globally. A geodesic between two points may cease to be the shortest route beyond a distance known as the cut locus. In spacetime, geodesics describe the paths of freely moving particles and light, governed by the spacetime metric.

Applications: In geodesy and navigation, geodesics on the Earth’s ellipsoid approximate shortest routes for long-distance travel. In physics, geodesics underpin general relativity, where curvature produced by mass-energy governs geodesic deviation and gravitational effects.

History: The concept originates in the study of surfaces and distances on the Earth, with roots in Gauss; the term geodesic comes from Greek words for earth and way, and the idea was extended by mathematicians into differential geometry.