The Geometry of Geodesics

The geometry of geodesics involves the shortest or most natural path between two points on a curved surface or space, like a sphere or a saddle shape. Imagine a great circle route on Earth (a geodesic) that airplanes follow to minimize distance. Geodesics are determined by the surface's curvature; on flat surfaces, they are straight lines, while on curved surfaces, they bend accordingly. In curved spacetime, like around planets or stars, geodesics describe the paths objects like planets or light follow under gravity, representing the natural movement dictated by the geometry of the space itself.