Geodesics

What is a geodesic?

A geodesic is the shortest path between two points on a surface — the generalization of a straight line to curved geometry. On a sphere, geodesics are great circles: circles whose center is the center of the sphere. The equator is a great circle. So is any line of longitude. But lines of latitude (except the equator) are not — they curve away from the shortest path.

Geodesics in general relativity

Einstein’s general relativity reinterprets gravity as the curvature of spacetime. Objects in free fall — planets orbiting stars, light bending around galaxies — follow geodesics through four-dimensional spacetime. They are not being “pulled” by a force; they are taking the straightest possible path through curved geometry. Mass tells spacetime how to curve; spacetime tells matter how to move.

Different surfaces, different geodesics

On a torus (donut shape), geodesics can wind around the tube, around the hole, or both — and the shortest path between two points depends on which way around you go. On a hyperbolic surface (saddle shape), geodesics diverge: parallel lines spread apart, and the sum of a triangle’s angles is less than 180°. Each geometry has its own character, and geodesics reveal it.

Euclidean vs. geodesic distance

The Euclidean distance between two points is the straight-line “through the surface” distance — like drilling a tunnel through the Earth. The geodesic distance is the shortest path along the surface — like flying from London to Tokyo. On a sphere, the geodesic distance is always longer than or equal to the Euclidean distance, and the difference grows as the points move farther apart (up to antipodal points, where the geodesic is half the circumference but the Euclidean distance is just the diameter).