a geodesic
/ jee-oh-DEH-sik /
A jet flying from London to Tokyo does not follow the straight line you would draw on a flat map; it arcs up over the Arctic. That arc is the shortest route across the curved Earth — a great-circle path. A geodesic is the generalisation of 'straight line' to any curved surface: it is the locally shortest path between nearby points, and equivalently the 'straightest possible' curve you can draw while staying on the surface.
There are two equivalent precise pictures, and seeing they agree is the heart of the idea. (1) Shortest-path view: a geodesic is a curve that locally minimises arc length — pin down two nearby points on the surface and pull a string taut between them along the surface, and it lies along a geodesic. (2) Straightest-curve view: a geodesic is a curve whose geodesic curvature is zero everywhere. That means its acceleration vector, as you traverse it at constant speed, points purely in the surface's normal direction with no component within the tangent plane — the curve never 'steers' sideways within the surface; any bending it shows is only the unavoidable bending forced by the surface itself. Equivalently, the velocity vector is parallel-transported along the curve: a geodesic carries its own direction forward as faithfully as the curved surface allows.
Geodesics are the surface's intrinsic notion of 'going straight,' and because they are defined by the first fundamental form alone they are an intrinsic concept — the ant can find them without leaving the surface. They are central everywhere curvature matters: light and free particles in Einstein's general relativity travel along geodesics of curved spacetime; the shortest shipping and flight routes are geodesics of the globe. Two honest cautions. First, geodesics are LOCALLY shortest, not always globally shortest: on a sphere a great circle is a geodesic, but going 'the long way round' it between two points is still a geodesic while obviously not the shortest route. Second, a geodesic is not a 'line of curvature' and is not generally a normal section; do not confuse 'straightest path' (geodesic) with 'principal bending direction' (line of curvature).
On a sphere, the geodesics are exactly the great circles — circles whose plane passes through the centre, like the equator or any meridian. The shortest flight from one city to another follows the great circle through both, which is why polar routes look curved on a flat map but are genuinely straightest on the globe. By contrast, a circle of latitude (other than the equator) is NOT a geodesic: walking it, you must constantly steer toward the nearer pole, so it has nonzero geodesic curvature.
On a sphere geodesics are great circles; ordinary latitude circles are not.
Geodesics are only LOCALLY shortest. Between two non-antipodal points on a sphere there are two great-circle arcs — the short way and the long way — and BOTH are geodesics, but only the short one minimises distance. 'Geodesic' means 'no sideways steering,' which guarantees shortest only over small enough pieces.