DIFFERENTIAL GEOMETRY OF SURFACES 187

nu · rv = a 1 ru·rv + b 1 rv·rv⇒–M = a 1 G 12 + b 1 G 22 ⇒b 1 = 0 (QM = 0, G 12 = 0)

Therefore, nruu
L
G
=–
11
(6.47)

Similarly,
nv = a 2 ru + b 2 rv⇒nv · ru = a 2 ru·ru + b 1 rv·ru⇒–M = a 2 G 11 + b 2 G 12
⇒a 2 = 0 (QG 12 = 0, M = 0)

nv · rv = a 2 ru·rv + b 2 rv·rv⇒–N = b 2 G 22 ⇒b 2 = – N

G 22

(Qa 2 = 0)

Therefore, nrvv=–

N
G 22 (6.48)
Using these expressions for nu and nv,K and H can be determined as follows:

rruuaGL ru rr ru r r

aL

G a

N

G

aN

G

*

11 11

*

22 22

= – = 1 – ⎛ , = – = 1 –

⎝

⎞

⎠

⎛

⎝

⎞

vvvv⎠ (6.49a)

G

aL

G

aL

G

11 ***uu uu G

11

2

11

2

= rr⋅ = 1 – rr = 1 – 11

⎛

⎝

⎞

⎠

⋅ ⎛

⎝

⎞

⎠ (6.49b)

G

aL

G

aN

G

aL

G

aN

G

12 ***uuG

11 22 11 22

= = 1 – 1 – = 1 – 1 – = 0rr⋅ rr 12

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⋅ ⎛

⎝

⎞

⎠

⎛

⎝

⎞

vv⎠ (6.49c)

G

aN

G

aN

G

22 *** G

22

2

22

2

= = 1 – rrvvvv⋅ = 1 –rr 22

⎛

⎝

⎞

⎠

⋅ ⎛

⎝

⎞

⎠ (6.49d)

L

aL

G L

aL

G M

aL

= –uu = – 1 – (^) uu = 1 – , = – u = – 1 – G u = 0

11 11

11
r n⋅ ⎛ rn r r rn
⎝
⎞
⎠
⋅ ⎛
⎝
⎞
⎠
⋅ ⎛
⎝
⎞
⎠
vv⋅
(6.49e)
N
aN
G
aN
G
*= – * = – 1 – = 1 – N
22 22
rnvvvv⋅ ⎛ rn
⎝
⎞
⎠
⋅ ⎛
⎝
⎞
⎠ (6.49f)
The principal curvatures at point P on S can now be determined as follows:
κκ 1
11

• 11
  11
  2


• 
  

  22

  
  


• 
  

  22
  22
  = =
  1–
  , = =
  1–
  L
  G
  L
  aL
  G G
  N
  G
  N
  aN
  G G
  ⎛
  ⎝
  ⎞
  ⎠
  ⎛
  ⎝
  ⎞
  ⎠
  (6.50)
  The Gaussian and mean curvatures of S* are given by
  K NL
  GG
  aL
  G
  aN
  G
  K
  a L
  G
  N
  G
  a NL
  GG
  K
  aH a K

  
  


• 
  

  = =
  1– 1–

  
  

  1 – + +

  
  

  1– 2 +
  2

  
  


• 
  

  2

  
  


• 
  

  22 11
  11 22
  11 22
  2
  11 22
  2
  κκ
  ⎛
  ⎝
  ⎞
  ⎠
  ⎛
  ⎝
  ⎞
  ⎠
  ⎛
  ⎝
  ⎞
  ⎠
  (6.51)
  H
  L
  G
  aL
  G
  N
  G
  aN
  G
  HKa
  aH a K

  
  


• =^1
  2
  ( + ) =^1
  2 1–

• 1–
  = +
  1– 2 +
  1 2
  11 11 22 22 2
  κκ ⎛
  ⎝
  ⎞
  ⎠
  ⎛
  ⎝
  ⎞
  ⎠
  ⎛
  ⎝
  ⎜
  ⎜
  ⎞
  ⎠
  ⎟
  ⎟ (6.52)