Geodesic Paths on Curved Surfaces
On a flat plane the shortest path is a straight line. On a curved surface, the shortest path is a geodesic — a great circle on a sphere, something stranger on a torus. Drag the endpoints to explore. Geodesics are the paths light and freely-falling objects follow through curved spacetime in general relativity.
What is a geodesic?
A geodesic is the generalization of a straight line to a curved surface: it is the locally shortest path between two points. On a sphere, every geodesic is a segment of a great circle — a circle whose plane passes through the center of the sphere. This is why airline routes curve across a flat map: the apparent curve is actually the shortest distance on the spherical Earth.
Curvature and stranger paths
On a torus (the surface of a doughnut) geodesics are far richer. The inner equator is shorter in circumference than the outer equator, so geodesics can wind around the tube multiple times before closing — or never close at all. On a hyperboloid (a saddle-shaped surface of negative Gaussian curvature) geodesics spread apart rather than converge, a direct contrast to the sphere where they always converge at the poles. On an ellipsoid, geodesics generalize great circles: the three principal ellipses are geodesics, but all other paths are quasi-periodic and non-closing in general.
Riemannian geometry
Geodesics are defined by the geodesic equation, which encodes how a tangent vector is parallel-transported along the path using the Christoffel symbols Γ — local measures of how the metric tensor changes. The metric tensor gij carries all information about distances and angles on the surface. This framework, developed by Riemann in 1854, is exactly the mathematics Einstein used 60 years later to describe gravity: in general relativity, massive objects curve the 4-dimensional spacetime metric, and freely-falling particles and light rays follow geodesics in that curved spacetime.
Navigation and general relativity
The practical importance of geodesics ranges from GPS satellite corrections (clocks run faster in weaker gravity, a geodesic effect) to the bending of starlight around the Sun first measured during the 1919 solar eclipse that confirmed general relativity. Any time you fly from New York to London and notice the route arcs north toward Greenland on a flat map, you are seeing a spherical geodesic. The mathematics of curved surfaces, developed as pure geometry in the 19th century, turned out to be the language of the universe.