geodesic curvature
/ jee-oh-DEH-sik /
Walk a curve painted on a hillside at a steady pace. Part of the turning you feel is unavoidable — the ground itself is curved and forces you to bob up and down. But part of the turning is your own doing: you are steering left or right WITHIN the surface, away from the straightest path. Geodesic curvature measures exactly that second part — how much the curve veers sideways inside the surface, ignoring the bending the surface forces on it.
Precisely, take a curve on a surface, traversed at unit speed, and look at its acceleration vector (the rate of change of its unit tangent). This acceleration splits into two perpendicular pieces: a component along the surface's normal direction, whose size is the normal curvature k_n, and a component lying IN the tangent plane, whose signed size is the geodesic curvature k_g. The normal part is the bending the surface compels; the geodesic part is the steering the curve does on its own. The two combine to give the curve's ordinary (extrinsic) curvature by kappa^2 = k_n^2 + k_g^2. The decisive feature: k_g depends only on the first fundamental form, so it is INTRINSIC — the ant on the surface can measure how much a path is steering without any view of the third dimension.
Geodesic curvature is the precise meaning of 'how far from straight' a surface curve is. A geodesic is exactly a curve with k_g = 0 everywhere — the straightest possible path, doing no steering of its own. Geodesic curvature is also the local ingredient of the Gauss-Bonnet theorem: integrating k_g around the boundary of a region, together with the integral of the Gaussian curvature over the region and the exterior angles at corners, yields 2*pi times the Euler characteristic. One caution worth holding: a curve can look very curved as a space curve yet have small or zero geodesic curvature, because most of its bending may be the forced normal part. On a sphere a great circle is genuinely curved in space (kappa = 1/R) but has k_g = 0 — all its bending is normal, none is steering, which is why it is a geodesic.
Consider a circle of latitude on a globe at latitude phi (say phi = 60 degrees N). As a curve it is genuinely curved, but how much of that is 'steering'? Its geodesic curvature is k_g = (1/R) tan phi. At the equator phi = 0, so k_g = 0 — the equator is a geodesic, doing no steering. As you go toward the pole phi grows and tan phi blows up, so k_g grows: a tight circle near the pole requires constant hard steering toward the pole to stay on its latitude, which is exactly why such circles are far from geodesics.
A latitude circle's geodesic curvature is (1/R)tan(phi): zero at the equator, large near the pole.
Geodesic curvature is intrinsic and measures only the IN-surface steering; it is not the same as a curve's ordinary curvature kappa, which also includes the forced normal part (kappa^2 = k_n^2 + k_g^2). A curve sharply curved in space can have k_g = 0 if all its bending is normal — that is precisely a geodesic.