Chapter 2: Curvatures of Curves and Surfaces via the Moving-Frame Method

In this chapter, we introduce the moving-frame method to study the differential properties of curves and surfaces. In particular, we obtain the infinitesimal displacement of the Frenet frame on a spatial curve in terms of curvature k and torsion τ and then the infinitesimal displacement of the Darboux frame on a surface curve in terms of geodesic curvature k_g, normal curvature k_n, and geodesic torsion τ_g.