Figure 3.10: Geometrical illustration of the second fundamental form Therefore PQ=r(u+du,u+dv)-r(u, u)=rudu+rudy+5(ruudu+2ruududu+rudu2)+HOT Thus, the projection of PQ onto N d=PQ N=(rudu +rudu)N+II 0, we get d==lI=s(Ldu+ 2Mdudv+ndu We want to observe in which situation d is positive and negative. When d=0 Ldu+2Mdudu+ Ndu= 0 d M±√(Md)2-D nda,2 M±√M2-LN (318) Figure 3.11:(a) Elliptic point;(b)Parabolic point;(c) Hyperbolic point If M2-LN < 0. there is no real root. That means there is no intersection between the surface and its tangent plane except at point P. P is called elliptic point(Figure 3.11(a)) If M2-LN=0, there is a double root. The surface intersects its tangent plane with one line du=-idu, which passes through point P. P is called parabolic point(Figure 3. 11(b)) If M2-LN >0, there are two roots. The surface intersects its tangent plane with two M+vAR-LNdv, which intersect at point P. P is called hyperbolic point (Figure 3.11(c))N P Q d Tp r=r(u,v) Figure 3.10: Geometrical illustration of the second fundamental form. Therefore PQ = r(u + du, v + dv) − r(u, v) = rudu + rvdv + 1 2 (ruudu2 + 2ruvdudv + rvvdv2 ) + H.O.T. Thus, the projection of PQ onto N d = PQ · N = (rudu + rvdv) · N + 1 2 II and since ru · N = rv · N = 0, we get d = 1 2 II = 1 2 (Ldu2 + 2Mdudv + Ndv2 ) We want to observe in which situation d is positive and negative. When d = 0 Ldu2 + 2Mdudv + Ndv2 = 0 Solve for du du = −M ± p (Mdv) 2 − LNdv2 L = −M ± √ M2 − LN L dv (3.18) N N N P P Tp P Tp Tp Figure 3.11: (a) Elliptic point; (b) Parabolic point; (c) Hyperbolic point. • If M2−LN < 0, there is no real root. That means there is no intersection between the surface and its tangent plane except at point P. P is called elliptic point (Figure 3.11(a)). • If M2 −LN = 0, there is a double root. The surface intersects its tangent plane with one line du = − M L dv, which passes through point P. P is called parabolic point (Figure 3.11(b)). • If M2 − LN > 0, there are two roots. The surface intersects its tangent plane with two lines du = −M± √ M2−LN L dv, which intersect at point P. P is called hyperbolic point (Figure 3.11(c)). 10