Geodesics

Stretch a string tight between two points on a curved surface — over a hill, around a globe — and it settles into the shortest path that stays on the surface. That path is a geodesic. Geodesics are the surface's answer to "straight line": the routes an ant would call straight, the tracks a rolling marble follows, the great-circle arcs an aeroplane flies.

There are three equivalent ways to say what a geodesic is, and it helps to hold all three at once:

Locally shortest. Between nearby points it minimises length along the surface.
Zero geodesic curvature. It never turns within the surface — it curves only as much as the surface forces it to.
Acceleration is purely normal. Its acceleration \ddot\sigma points straight out along the normal \mathbf{n}, with no component in the tangent plane. A geodesic feels no sideways push.

From "shortest" to an equation

The first description — shortest path — turns this into a problem for the Euler–Lagrange equation. Recall that the length of a curve t\mapsto(u(t),v(t)) is measured by the first fundamental form:

L[u,v] = \int \sqrt{E\,\dot u^2 + 2F\,\dot u\,\dot v + G\,\dot v^2}\;dt.

A geodesic is a curve that makes this length functional stationary — exactly the setup the Euler–Lagrange equation was built for, now with two unknown functions u(t) and v(t). (In practice one extremises the energy \int (E\dot u^2 + 2F\dot u\dot v + G\dot v^2)\,dt instead — same geodesics, no awkward square root to differentiate.)

Grinding the Euler–Lagrange equations through produces the geodesic equations. Their compact form uses the Christoffel symbols \Gamma^k_{ij}, bookkeeping quantities assembled purely from E, F, G and their derivatives:

\ddot\gamma^k + \Gamma^k_{ij}\,\dot\gamma^i\dot\gamma^j = 0, \qquad \gamma = (u, v).

You do not need to compute these by hand here — the point is what they are made of. The Christoffel symbols come only from the metric E, F, G, so the geodesic equation is intrinsic: geodesics depend on nothing but the first fundamental form. An ant could work them out without ever seeing the third dimension.

Three surfaces, three geodesics

The plane. With E=G=1,\ F=0 all Christoffel symbols vanish, so \ddot u = \ddot v = 0 — constant velocity. The geodesics are exactly the straight lines, and the geodesic between two points has the ordinary straight-line length. Reassuringly, the machinery rediscovers what we already knew.

The sphere. The geodesics are the great circles — the equator, the meridians of longitude, and every circle whose plane passes through the centre. The shortest flight from London to Tokyo follows a great circle, which is why it curves up over the Arctic on a flat map. Small circles (like most lines of latitude) are not geodesics: to stay on them you would have to keep steering sideways.

The cylinder. The geodesics are helices (plus the straight vertical lines and the horizontal circles as special cases). Here is the loveliest argument in the whole subject: a cylinder is developable — you can unroll it onto a flat sheet without stretching. Because unrolling is an isometry, it preserves lengths, so it preserves shortest paths. On the flat sheet the shortest path is a straight line; roll the sheet back up and that straight line becomes a helix. Bending doesn't change the metric, and the metric is all geodesics know about.

Unroll a cylinder and watch the helix straighten

Below is a cylinder cut open and rolled flat — its surface is now an ordinary rectangle (the left and right dashed edges are glued back together to re-form the cylinder). The straight coloured line is the geodesic between the two marked points. Drag the sliders to move the far endpoint: however far around (\text{arc}) and up (\text{height}) it sits, the geodesic on the flat sheet is a straight line, with length \sqrt{\text{arc}^2 + \text{height}^2} by Pythagoras. Roll the sheet back up and this straight line coils into a helix — the shortest path on the cylinder.

Vignettes

On a flat wall-map the route looks absurd — a great northern arc when a straight ruler line across the Pacific seems shorter. But the map is lying, because it flattened a sphere (and Theorema Egregium promised us it must distort). On the actual globe the shortest path between two cities is the great-circle arc, and for two northern-hemisphere cities that arc bends poleward.

Airlines fly great circles to burn less fuel, so long-haul routes really do hug the top of the world — planes from North America to East Asia routinely skirt the Arctic, and Southern-Hemisphere routes bend toward Antarctica. The "straight line" your instinct wants is straight only on a distorted piece of paper; the geodesic is straight on the Earth.

A geodesic is only locally shortest, not globally. Take two nearby cities on the equator: the short way round is a geodesic — but so is the long way, all the way round the other side of the planet. Both are arcs of the same great circle, both satisfy the geodesic equation, yet only one is the genuine shortest path. Solving the geodesic equation gives you a candidate straight path; it does not promise it is the shortest of all. (This is the same "stationary, not necessarily minimal" caution as for the Euler–Lagrange equation itself.)

A second reminder: geodesics are intrinsic. They are determined entirely by the first fundamental form E, F, G — bend a surface without stretching it and its geodesics come along unchanged. That is exactly why unrolling the cylinder works: the flat sheet and the cylinder share a metric, so they share geodesics, straight line and helix being the very same path seen two ways.