us⊥n us1nds]nv us]nacos Figure 3.7: Ci Ircular cone -u sin a cos a cos ver-u sin a cos a sin vey t usin" aez At the origin n=0 Therefore, the apex of the cone is a singular point 3.6 Second fundamental form II(curvature) k Figure 3. 8: Definition of normal curvature In order to quantify the curvatures of a surface S, we consider a curve C on S which passes through point P as shown in Figure 3.8. t is the unit tangent vector and n is the unit normal vector of the curve C at point P ds where kn is the normal curvature vector normal to the surface, kg is the geodesic curvature vector tangent to the surface, and k= kn is the curvature vector of the curve C at point P kin is called the normal curvature of the surface at p in the direction tp α v u y x z usinαcosv usinαsinv singular α u usinα Figure 3.7: Circular cone. = −u sinα cos α cos vex − u sin α cos α sin vey + u sin2 αez At the origin n = 0, ru × rv = 0 Therefore, the apex of the cone is a singular point. 3.6 Second fundamental form II (curvature) S P N kg k kn n C t Figure 3.8: Definition of normal curvature In order to quantify the curvatures of a surface S, we consider a curve C on S which passes through point P as shown in Figure 3.8. t is the unit tangent vector and n is the unit normal vector of the curve C at point P. dt ds = κn = kn + kg (3.11) kn = κnN (3.12) where kn is the normal curvature vector normal to the surface, kg is the geodesic curvature vector tangent to the surface, and k = κn is the curvature vector of the curve C at point P. κn is called the normal curvature of the surface at P in the direction t. 8