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)