Graphics Reference
In-Depth Information
tant a role as the functions E, F, and G. In particular, let us show that the derivative
of the Gauss map can be described in terms of these functions. First of all, since
n
u
(u,v) and
n
v
(u,v) lie in the tangent space at F(u,v) (
n
•
n
= 1 implies
n
u
•
n
=
n
v
•
n
= 0), we can express the vectors in terms of the local basis, namely,
n
n
=
c
FF
FF
+
c
u
11
u
12
v
=
c
+
c
(9.48)
v
21
u
22
v
for some constants c
ij
. Equations (9.46) and (9.48) imply that
()
=
(
)
(
)
D
n
g
cucv
¢ +
¢
F
+
cucv
¢ +
¢
F
(9.49)
11
21
u
12
22
v
and that the matrix for D
n
with respect to the basis (F
u
,F
v
) is
cc
cc
Ê
Ë
ˆ
¯
.
11
12
(9.50)
21
22
But from equation (9.48) and the definition of the coefficients of the first fundamen-
tal form we get
-=
L
n
n
n
n
∑
F
F
F
F
=
c
E
+
c
F
uu
11
12
-= ∑ =
M
c
F
+
c
G
u
v
11
12
-= ∑ =
M
c
E
+
c
F
v
u
21
22
-= ∑
N
=
c
F
+
c
G
.
(9.51)
v
v
21
22
We can express equations (9.51) in matrix form as
LM
MN
cc
cc
EF
FG
-
Ê
Ë
ˆ
¯
=
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
11
12
.
(9.52)
21
22
This is easily solved for the matrix (c
ij
) to establish that
MF
-
-
LG
c
=
,
(9.53a)
11
2
EG
F
LF
-
-
ME
c
=
,
(9.53b)
12
2
EG
F
NF
-
-
MG
c
=
,
(9.53c)
21
2
EG
F
MF
-
-
NE
c
=
.
(9.53d)
22
2
EG
F
