geodesically

Geodesically is an adverb describing actions, properties, or constructions that relate to geodesics. In differential geometry, a geodesic is a curve that locally minimizes length between nearby points and satisfies the geodesic equation; it can be parameterized by arc length. Geodesics generalize straight lines to curved spaces and, in physics, describe the motion of free particles in the absence of external forces. In general relativity, light and matter follow geodesics of spacetime determined by its metric.

A space or a subset may be described as geodesic or geodesically structured in several senses. A geodesic metric space is one in which the distance between any two points is realized by a geodesic segment. A set is geodesically convex if every geodesic joining two points in the set lies entirely within it. A manifold is geodesically complete if all geodesics can be extended for all parameter values, a property related to the global geometry of the space through results such as the Hopf–Rinow theorem.

Examples help anchor the concept. In Euclidean space, geodesics are straight lines; on a sphere, they are great circles. The term derives from geodesy, the science of measuring the Earth, where shortest paths on the globe are modeled by geodesics. When a curve or construction is described as geodesic or geodesically parameterized, it reflects alignment with the geodesic flow, often with arc-length as the natural parameter.