DIFFERENTIAL GEOMETRY OF SURFACES 181

G 11 = ru·ru = 1 + (3u^2 – 3v^2 )^2 , G 12 = ru·rv = – 6uv (3u^2 – 3v^2 ),

G 22 = rv·rv = 1 + 36u^2 v^2

The surface normal is determined as

n

rr

rr

=

| |

=

(–3 + 3 , 6 , 1)

1 + 9 + 9 + 18

22

44 22

u

u

uu

uu

×

×

v

v

vv

vv

Therefore,L
u
uu

M

uu

= uu = uv

6

1 + 9 + 9 + 18

, = =

–6

44 22 1 + 9 + 9 + 1844 22

nr⋅⋅nr

vv

v

vv

N

u

uu

= =

–6

1 + 9 + 9 + 1844 22

nr⋅ vv

vv

From Eq. (6.36) we have the Gaussian and mean curvatures

K

LN M

GG G

u

uu

=

=

–36( + )

(1 + 9 + 9 + 18 )

2

(^112212)
2
22
44 222
v
vv
H
GN G L GM
GG G
uu u
uu

• 
  

  – 2
  2( – )

  
  

  –27 + 54 + 81
  (1 + 9 + 9 + 18 )
  11 22 12
  (^112212)
  2
  532 4
  44 23/2
  vv
  vv
  The monkey saddle and its curvatures are shown in Figures 6.13. Maximum and minimum normal
  curvatures can be determined using Eqs. (6.36) with expressions of Gaussian and mean curvatures
  derived above.
  Foregoing discussion dealt with the differential properties of surfaces that included the tangent
  plane and normal at a point, the first and second fundamental matrices, and principal (maximum and
  minimum normal), Gaussian and mean curvatures. Such properties are mainly studied for composite
  fitting of surface patches at their common boundaries to achieve slope and/or curvature continuity.
  Before generic design of surface patches is ventured into in Chapter 7, a notion about some specific
  patches is provided below. A user may not need to specify the slope or data point information directly
  for their design. However, the input sought would be indirect, more like in terms of a pair of curves
  (for instance Ferguson, Bézier or spline curves), or a curve and a straight line.

  
  

6.6 Developable and Ruled Surfaces

Developable surfaces can be unfolded or developed onto a plane without stretching or tearing (Figure
6.14a). Such surfaces are useful in sheet metal industry for making drums, conical funnels, convergent
or divergent nozzles, ducts for air conditioning, shoes, tailoring shirts and pants, automobile upholstery,
door panels, windshield, shipbuilding, fiber reinforced plastic (FRP) panels for aircraft wings, and
many other applications. Hence, the design of developable surfaces cannot be ignored. Cylindrical
and conic patches are well-known examples. An interesting note about a developable surface is that
at every point on the surface, the Gaussian curvature K is zero. Thus, they are known as singly curved
surfaces,since one of their principal curvatures is zero.
From Eq. (6.37), the Gaussian curvature K is zero if either κminorκmax, or both are zero. Since
GG 11 22 – > 0G 122 as shown earlier, K = 0 implies that LN–M^2 = 0. As discussed in Section 6.4, this