DIFFERENTIAL GEOMETRY OF SURFACES 173

The two curves are orthogonal to each other if

∂

∂

∂

∂

⎛

⎝

⎜

⎞

⎠

⎟⋅

∂

∂

∂

∂

⎛

⎝

⎜

⎞

⎠

⎟

rr rr(, )

+

( , ) ( , )

+

(, )

= 0

11 22

u

u

du

dt

ud

dt

u

u

du

dt

ud

dt

vv

v

vvv

v

v

or G

du

dt

du

dt

G

du

dt

d

dt

du

dt

d

dt

G

d

dt

d

(^11121212212212) dt

• 
  
  
  
  • 
    
    
    
    • 
      

      = 0
      ⎛ vvvv
      ⎝
      ⎞
      ⎠ (6.13)
      If the two curves coincide, respectively, with u and viso-parametric curves of the surface, then we
      may regard u≡t 1 and v≡t 2 for which the dot product in Eq. (6.12) becomes
      tt
      rr
      rr
      rr
      rr
      rr rr
      12
      11 22
      12
      11 22
      = cos =
      ( , ) ( , )
      = =
      | || |

      
      

      ⋅
      ∂
      ∂
      ⋅
      ∂
      ∂ ⋅⋅
      ⋅⋅
      θ
      u
      u
      G
      u
      G
      G
      GG
      u
      u
      u
      uu
      vv
      v v
      v
      v
      vv
      (6.14)
      Thus, the iso-parametric curves are orthogonal if G 12 = 0. To compute the area of a surface patch
      r (u,v), let a small patch on the surface be formed by the curves between u = u 0 ,u = u 0 + du,v = v 0
      andv = v 0 + dv. The four corners of this patch are r(u 0 ,v 0 ),r(u 0 + du,v 0 ),r(u 0 ,v 0 + dv) and
      r(u 0 + du,v 0 + dv) as shown in Fig. 6.8. The infinitesimal area dA is approximated by
      d A = | rruudu ××vvdvvvv | = | | rrdu d = G G 11 22 – G 122 dud = | |Gdu d
      Therefore, the area of the patch is given by
      Adud = | |
      ∫Domain
      G v (6.15)
      r(u 0 ,v 0 + dv)
      r(u 0 ,v 0 )
      ru dv
      dA = | ru×rv | dudv
      r(u 0 + du,v 0 )
      ru dv
      Figure 6.8 Infinitesimal area on the surface

      
      
    
    
    
    
  
  
  
  

6.3 Deviation of the Surface from the Tangent Plane: Second Fundamental Matrix

In Figure 6.9, let R(r(u 0 + du,v 0 + dv)) be a point on the surface a small distance away from
P(r(u 0 ,v 0 )). The deviation d of R from P along the normal n at P may be written as

d = [ r(u 0 + du,v 0 + dv)–r(u 0 ,v 0 )] ·n