Umbilics and Lines of Curvature

Abstract

An umbilic is a point on a surface where all normal curvatures are equal in all directions, and hence principal directions are indeterminate. Thus the orthogonal net of lines of curvature, which is described in Sect. 3.4, becomes singular at an umbilic. An obvious example of a surface consisting entirely of umbilical points is the sphere. Actually, spheres and planes are the only surfaces all of whose points are umbilics. An obvious example of a surface consisting entirely of umbilical points is the sphere. Actually, spheres and planes are the only surfaces all of whose points are umbilics. The number of umbilics on a surface is often finite and they are isolated ¹ [165, 412]. Umbilics have generic features and may act as fingerprints for shape recognition. At an umbilic, the directions of principal curvature can no longer be evaluated by second order derivatives and higher order derivatives are necessary to compute the lines of curvature near the umbilic. Monge (1746-1818), who with Gauss can be considered as the founder of differential geometry of curves and surfaces, first computed the lines of curvature of the ellipsoid (1796) which has four umbilics [320].