Geodesics

Zero geodesic curvature: the surface's straight lines and shortest paths.

The idea

A geodesic is the surface analogue of a straight line: a curve that turns no more than the surface forces it to.

Parametrize a curve on a surface by arc length and split its acceleration $\ddot{\gamma}$ into a component normal to the surface and a component tangent to it. The normal component is compulsory — it is what keeps the curve on the surface — while the tangential component is the curve's own turning within the surface. The signed length of the tangential component is the geodesic curvature $\kappa_{g}$, and a geodesic is a curve with $\kappa_{g} = 0$ everywhere. In the plane nothing is compulsory, so the condition reduces to $\ddot{\gamma} = 0$ and the geodesics are the ordinary straight lines.

Although this splitting refers to the surrounding space, $\kappa_{g}$ is intrinsic: it can be computed from the first fundamental form alone. Bending a surface without stretching it therefore carries geodesics to geodesics — roll a flat sheet into a cylinder and its straight lines become helices, still geodesics, because no internal measurement has changed. Geodesics are also the locally shortest paths: a sufficiently short arc of a geodesic is the shortest curve on the surface between its endpoints.

Ways to work on it

• Walkthrough. Geodesic curvature, and why great circles are the sphere's geodesics.
• Practice. Use Clairaut's relation on a surface of revolution.
• Hardest. Find a geodesic's turning parallel from its Clairaut constant.

Not sure where to start? Take the ten-question placement test.