geodesica

Geodesica is a term used in several languages to denote geodesic concepts in geometry and physics. In mathematics, a geodesic on a smooth manifold with a metric is a curve whose tangent vector remains parallel to itself along the curve, equivalently a curve that locally minimizes length or energy. On a sphere, geodesics are great circles.

In physics, geodesics describe the motion of free particles in curved spacetime. Timelike geodesics represent possible worldlines of massive bodies, while null geodesics describe the paths of light. The geodesic equation, involving Christoffel symbols, governs how trajectories bend in response to curvature; nearby geodesics diverge or converge as described by the geodesic deviation equation.

Geodesics are used in practical computations: on the Earth’s surface, the shortest route between two points is the geodesic path, approximated by great-circle distance in navigation and geographic information systems. In computer graphics and geometry processing, geodesic paths on meshes are computed for texture transfer, path finding, and shape analysis.

Originating from Greek words for earth (geo) and path (hodos), the concept of geodesics is central to differential geometry and general relativity, linking straight lines in flat space to their curved-space generalizations.