from https://en.wikipedia.org/wiki/Geodesic

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In geometry, a **geodesic** (/ˌdʒiː.əˈdɛsɪk, -oʊ-, -ˈdiːsɪk, -zɪk/)^{[1]}^{[2]} is commonly a curve representing in some sense the shortest^{[a]} path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a „straight line„.

The noun *geodesic*^{[b]} and the adjective *geodetic*^{[c]} come from *geodesy*, the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth’s surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph.

In a Riemannian manifold or submanifold, geodesics are characterised by the property of having vanishing geodesic curvature. More generally, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion.

Geodesics are of particular importance in general relativity. Timelike geodesics in general relativity describe the motion of free falling test particles.