geodesic
/ jee-oh-DESS-ik /
On a flat plane the shortest route between two points is a straight line, and a straight line is also the path you trace if you walk without ever turning the wheel. On a curved surface those two ideas — shortest and straightest — still point to the same special curves, called geodesics. An airliner flying the shortest route from one city to another follows a great circle that on a flat map looks bizarrely bowed; that great circle is a geodesic of the sphere.
There are two equivalent ways to pin a geodesic down, and it is worth holding both. The 'straightest' definition: a geodesic is a curve that parallel-transports its own tangent vector — it never accelerates sideways within the surface, so its covariant acceleration is zero. This gives the geodesic equation d^2 x^k/ds^2 + Gamma^k_{ij} (dx^i/ds)(dx^j/ds) = 0, where s is arc length and the Christoffel terms account for the curving of the space. The 'shortest' definition: a geodesic is a stationary point of the length functional (integral of ds) between its endpoints — applying the Euler-Lagrange equation of the calculus of variations to that length integral reproduces exactly the same geodesic equation. So geodesics are simultaneously the locally length-extremizing curves and the no-turning curves, a beautiful coincidence of variational and differential viewpoints.
Geodesics are the curved-space replacement for straight lines and they are everywhere: great circles on the globe (the basis of long-haul flight planning), the shortest cable routes draped over terrain, the paths of light and free-falling bodies in general relativity (a planet orbits the Sun because it follows a geodesic of curved spacetime — gravity is geometry, not a force), and the optimal trajectories in many robotics and graphics problems. Two honest cautions: 'geodesic' means locally extremal, not necessarily globally shortest — on a sphere the long way around a great circle is still a geodesic but is the longer route between the two points; and a geodesic is generally only the SHORTEST path between sufficiently close points, while between far points it may be a saddle of the length, or one of several competing geodesics.
On a sphere the geodesics are exactly the great circles — circles whose center coincides with the sphere's center, like the equator or any line of longitude. A flight from New York to Tokyo follows the great circle that arcs up near the Arctic, which is shorter than the straight-looking line on a Mercator map even though the map makes it look like a detour.
Great circles are the sphere's geodesics — the 'bowed' flight path on a flat map is actually the shortest route.
A geodesic is locally straightest and locally shortest, but not always globally shortest: the major arc of a great circle is a perfectly good geodesic yet the longer way around. And straightest (zero covariant acceleration) is the more fundamental definition, since it still makes sense even when the metric has indefinite signature, as in relativity.