Geodesics: Explained with Examples, A Complete Student Guide

A geodesic is the straightest possible path on a curved surface: the route a surface-bound ant would call straight, and usually the shortest route between nearby points. On a plane geodesics are straight lines, on a sphere they are great circles like the equator, and in general they are curves whose acceleration has no component along the surface.

Geodesics generalise straight lines to curved spaces. Airlines fly great-circle arcs to minimise distance, GPS computes shortest paths on the Earth's curved surface, and in general relativity planets and light follow geodesics of curved spacetime, which is why gravity is geometry rather than a force.

Key Concepts

• Geodesic: a curve with vanishing tangential acceleration; locally length-minimising.
• Geodesic curvature: the within-surface part of curvature; zero exactly for geodesics.
• Great circle: sphere intersected with a plane through its centre; precisely the sphere's geodesics.
• Geodesic equations: second-order ODEs in surface coordinates whose solutions are geodesics.
• Cut locus: beyond some distance a geodesic stops being shortest; on a sphere, past the antipodal point.

Formulas

• Test: gamma is a geodesic iff gamma'' is parallel to the surface normal.
• Sphere: shortest distance between two points is R times the angle between their position vectors.
• Plane: with E = G = 1, F = 0, the geodesic equations give u'' = 0, v'' = 0: straight lines.

Worked Example

Show gamma(s) = (cos s, sin s, 0) is a geodesic of the unit sphere; find the geodesic distance from (1,0,0) to (0,1,0).

1. gamma lies on the unit sphere and is unit-speed since |gamma'| = 1.
2. gamma''(s) = (-cos s, -sin s, 0) = -gamma(s), which is the sphere's normal direction.
3. Zero tangential acceleration means gamma is a geodesic: the equator, a great circle.
4. gamma(0) = (1,0,0), gamma(pi/2) = (0,1,0), so the distance is pi/2 ≈ 1.5708.
5. The 45-degree latitude arc between the same longitudes is longer and not a geodesic.

Common Misconceptions

• Geodesics are only locally shortest. The long way round a great circle is a geodesic but not the shortest path.
• Latitude parallels are not geodesics (except the equator); they curve within the surface.
• Straight on a curved surface need not look straight: great-circle routes look curved on flat maps.

Real-World Applications

• Aviation: long-haul flights follow great-circle arcs, e.g. Delhi to New York via the Arctic.
• GPS: Earth distances follow geodesics of the reference ellipsoid.
• Relativity: free fall and light follow spacetime geodesics, explaining gravitational lensing.

A geodesic has no tangential acceleration: the natural meaning of straight on a curved surface. The equator is one on the unit sphere, giving distance pi/2 between (1,0,0) and (0,1,0). From flight paths to falling planets, geodesics are how the straightest routes are travelled.