# Geodesic

• a geodesic triangle on the sphere. the geodesics are great circle arcs.

in differential geometry, a geodesic (k/[1][2]) is a curve representing in some sense the shortest path between two points in a surface, or more generally in a riemannian manifold. it is a generalization of the notion of a "straight line" to a more general setting.

the term "geodesic" comes from geodesy, the science of measuring the size and shape of earth. 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. the term has been generalized to include measurements in much more general 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.

• introduction
• metric geometry
• riemannian geometry
• affine geodesics
• computational methods
• applications