géodésique

A géodésique, in the context of geometry, is the shortest path between two points on a given surface or in a given space. Imagine stretching a rubber band between two points on a flat sheet of paper; the path the rubber band takes is a straight line, which is the géodésique on that flat surface. On a curved surface, like the surface of a sphere, the géodésique is not a straight line in the conventional sense. Instead, it follows the curvature of the surface. For example, the shortest distance between two cities on Earth is not a straight line through the Earth, but a segment of a great circle on the Earth's surface.

The concept of a géodésique is fundamental in differential geometry and has applications in various fields, including physics, cartography, and navigation. In general relativity, géodésiques describe the paths of objects moving freely under the influence of gravity. These paths are not necessarily straight lines in spacetime because gravity is understood as the curvature of spacetime itself. Therefore, objects, including light, follow géodésiques through this curved spacetime. The mathematical definition of a géodésique involves solving differential equations that describe how the path changes direction infinitesimally to maintain the shortest distance property.