Geodesic curvature

Remarks

Since $\alpha'$, $N \times \alpha'$ and $N$ form an orthonormal basis and $\ip{\alpha''}{\alpha'} = 0$, one has $\alpha'' = \kappa_g \, N \times \alpha' + \kappa_n N$. So $\kappa^2 = \kappa_g^2 + \kappa_n^2$, where $\kappa$ is the curvature of $\alpha$ as a space curve. Example: on the unit sphere with the outer normal $N(p) = p$, a great circle $\alpha$ satisfies $\alpha'' = -\alpha$, which is normal to the sphere, so $\kappa_g = 0$. A circle of latitude at polar angle $\varphi_0 \in (0, \pi)$ has $\kappa = 1 / \sin \varphi_0$ and $\abs{\kappa_g} = \abs{\cot \varphi_0}$. Non-example: for a curve that is not of unit speed, the formula above is not the geodesic curvature; one must first reparametrise by arc length. For a curve in the $xy$-plane with $N = (0, 0, 1)$, the geodesic curvature is the signed curvature of the plane curve.