Meusnier's Theorem: All curves lying on a surface s passing through a given point pE S with the same tangent line have the same normal curvature at this point Since n.t=0. differentiate w r.t. t N′.t+N,t dt dr dN d Recoginizing that ds.ds=dx2+dy 2+dz2=dr, dr, we can rewrite Equation 3.13 as ter of center of curvature Figure 3. 9: Definition of positive normal:(a)kn. N=Kn;(b)kn.N=-Kn II = -dr dN=-(rudu+rudu)(Nudu Ndu Dudu+ nd (314) r MEN.r N=N,r (315) Therefore the normal curvature is given by ⅠIL+2MA+NX2 I E+2FA+GA2 (3.16) Suppose P is a point on a surface and Q is a point in the neighborhood of P, as in Figure 3.10. Taylor's expansion gives r(u+du, v+du)=ru, v)+rudu +r,du+fruudu-+ 2ruududv +roudu )+HOT( 3.17)Meusnier’s Theorem : All curves lying on a surface S passing through a given point p ∈ S with the same tangent line have the same normal curvature at this point. Since N · t = 0, differentiate w.r.t. s d ds(N · t) = N0 · t + N · t 0 dt ds · N = −t · dN ds = − dr ds · dN ds (3.13) Recoginizing that ds · ds = dx2 + dy2 + dz2 = dr · dr, we can rewrite Equation 3.13 as: dt ds · N = = − dr · dN dr · dr while dt ds · N = κn · N ≡ κn center of curvature center of curvature N N P P (a) (b) Figure 3.9: Definition of positive normal: (a) κn · N = κn; (b) κn · N = −κn. II = −dr · dN = −(rudu + rvdv) · (Nudu + Nvdv) = Ldu2 + 2Mdudv + Ndv2 (3.14) where L = N · ruu, M = N · ruv, N = N · rvv (3.15) Therefore the normal curvature is given by κn = II I = L + 2Mλ + Nλ 2 E + 2Fλ + Gλ2 (3.16) where λ = dv du . Suppose P is a point on a surface and Q is a point in the neighborhood of P, as in Figure 3.10. Taylor’s expansion gives r(u + du, v + dv) = r(u, v) + rudu + rvdv + 1 2 (ruudu2 + 2ruvdudv + rvvdv2 ) + H.O.T. (3.17) 9