Graphics Reference
In-Depth Information
Further, we abbreviate
n
(F(u,v)) to
n
(u,v) and define
∂
(
)
∂
(
)
∂
∂
n
n
o
F
∂
∂
n
n
o
F
(
)
=
(
)
∫
(
)
(
)
=
(
)
∫
(
)
n
uv
,
uv
,
uv
,
and
n
uv
,
uv
,
uv
,
.
u
v
u
∂
u
v
∂
v
The chain rule implies that
(
()
)
(
¢
()
)
=
(
)
¢
()
+
(
)
¢
()
Dt
n
g
g
t
n
uvut
,
n
uvvt
,
.
(9.46)
u
v
It follows from equations (9.46) and (9.35) that
()
=-
() ()
∑¢
Q
g
D
np
nn
FF
2
g
g
II
(
)
∑
(
)
=-
uv
¢+
¢
uv
¢ +
¢
u
v
u
v
2
=¢
Lu
+
2
Mu v
¢ ¢ +¢
Nv
,
(9.47)
where, using the identities that one gets when the known equations
n
• F
u
= 0 =
n
• F
v
are differentiated,
L
M
=-
n
∑
F
=
n
∑
F
,
uu
u
-∑
n
F
=∑
n
F
=∑
n
F
-∑
n
F
,
and
v
u
uv
vu
u
v
N
=-
n
∑
F
=
n
∑
F
.
v
v
vv
Definition.
The functions L, M, and N are called the
coefficients of the second
fundamental form
.
Note that the matrix
LM
MN
Ê
Ë
ˆ
¯
is just the matrix of the symmetric bilinear map associated to the quadratic form Q
II
with respect to the basis consisting of the vectors F
u
and F
v
and LN - M
2
is the dis-
criminant of Q
II
with respect to that basis. In terms of the shape operator S we have
(
∑
LS
M
=
FF
FF
,
u
u
()
∑=
(
∑
=
S
S
FF
,
and
v
u
u
v
(
∑
NS
=
FF
.
v
v
Note also that equation (9.47) expresses the Dupin indicatrix in terms of the functions
L, M, and N. Asymptotic lines are defined by the equation
2
2
()
+
()()
+
()
=
Lu
2
Mu v
Nv
0.
Next, compare Equation (9.47) for the second fundamental form with Equation
(9.39) for the first fundamental form. The functions L, M, and N play just as impor-
