Figure 3.6: The normal to the point on a surface A regular (ordinary) point P on the surface is defined as one for which ru xr+0. A point point P the vectors ru and r, do not vanish and have different directions equires that at that here ru x ru=0 is called a singular point. The condition ru x rufore Example: Elliptic Paraboloid r(u,v)=(u+v,u-U,u2+u) (1,1,2u) (1,-1,20) (+)er+2(u-)ey-2e≠ (x+u)2+( 2√2u2+202+1>0→ Regular! (2(+v),2(u-t),-2) 2V2u2+2211 at(t,v)=(0,0),N=(0,0,-1) Example: Circular Cone r(u, u)=(usin a cos U, u sin a sin v, u cos a), see Figure 3.7 (sin a cos v, sin a sin v, cos a) 0) u sin a sin uu sin a cosu 0x y z Tp N ru rv Figure 3.6: The normal to the point on a surface. A regular (ordinary) point P on the surface is defined as one for which ru ×rv 6= 0. A point where ru × rv = 0 is called a singular point. The condition ru × rv 6= 0 requires that at that point P the vectors ru and rv do not vanish and have different directions. Example: Elliptic Paraboloid r(u, v) = (u + v, u − v, u 2 + v 2 ) ru = (1, 1, 2u) rv = (1, −1, 2v) ru × rv =        ex ey ez 1 1 2u 1 −1 2v        = 2(u + v)ex + 2(u − v)ey − 2ez 6= 0 |ru × rv| = 2 q (u + v) 2 + (u − v) 2 + 1 = 2 p 2u 2 + 2v 2 + 1 > 0 ⇒ Regular ! N = (2(u + v), 2(u − v), −2) 2 √ 2u 2 + 2v 2 + 1 = (u + v, u − v, −1) √ 2u 2 + 2v 2 + 1 at (u, v) = (0, 0), N = (0, 0, −1) Example: Circular Cone r(u, v) = (u sin α cos v, u sin α sin v, u cos α), see Figure 3.7 ru = (sin α cos v,sin α sin v, cos α) rv = (−u sin αsinv, u sin αsinv, 0) ru × rv =        ex ey ez sinα cos v sin α sin v cos α −u sinα sin v u sin α cos v 0        7