géodésiques

A geodesic is the shortest path between two points on a curved surface or in a curved space. Imagine stretching a string between two points on a globe; the path the string takes is a geodesic. On a flat surface, a geodesic is simply a straight line. However, on curved surfaces like spheres or in more complex geometries, geodesics generally appear as curved lines to an observer in a flat space.

The concept of geodesics is fundamental in differential geometry and is crucial in understanding the geometry of curved spaces. In Euclidean geometry, straight lines are the simplest geodesics. In non-Euclidean geometries, such as Riemannian geometry, geodesics are defined by a set of differential equations.

In physics, particularly in Einstein's theory of general relativity, geodesics play a vital role in describing the motion of objects under the influence of gravity. Massive objects warp spacetime, and other objects move along the geodesics of this curved spacetime. This means that what we perceive as gravitational force is actually objects following the straightest possible paths in a curved space.

The term "geodesic" also has applications in fields outside of pure mathematics and physics, such as computer graphics and robotics, where it is used to find efficient paths on surfaces. Understanding geodesics helps us to quantify and describe shortest paths in various geometric settings.