Curves on a Surface

A curve on a surface such that the tangent line to it at every point is directed along a principal direction is called a line of curvature. A point at which principal curvatures of the surface are equal that is called an umbillic or naval point. At the umbillic point, the normal curvature is the same in every direction. At each point of a surface that is non-umbillic, there exist two mutually orthogonal directions for which the normal curvature attains its extreme values. The Rodrigues formula is characteristic for lines of curvature. The chapter concludes that the lines of curvature on a minimal surface form an isometric system.