Graphics Reference
In-Depth Information
We know that the length of the curve g is computed from the lengths of its tangent
vectors. If
()
=
(
()
()
()
)
m
t
m
t
,
m
t
,...,
m
t
,
1
2
n
then the chain rule implies that
n
∂
∂
F
Â
¢
()
=
(
()
)
¢
()
g
t
mm
t
t
.
(9.35)
i
u
i
i
=
1
Definition.
The
metric coefficients
g
ij
of the parameterization F and their deter-
minant g are the functions defined by
∂
∂
FF
∂
∂
=
()
g
=∑
uu
and
g
det g
ij
.
ij
i
j
Using the notation of metric coefficients, it follows from Equation (9.35) that
n
Â
¢
()
∑
()
=
(
()
)
¢
()
¢
()
g
t
g
t
g
mmm
t
t
t
.
(9.36)
ij
i
j
ij
,1
=
In other words, the length of the curve g(t) in the manifold is just the length of the
corresponding ordinary curve m(t) in
R
n
modified by the metric coefficients that
depend on the parameterization and the Riemannian metric.
A vector
v
in the tangent plane at a point
p
of a manifold is just the tangent vector
to some curve in the manifold. Therefore, if
v
=g¢(0), then we can use equation (9.36)
to rephrase the definition of the fundamental form Q
I
(
v
) at
p
as
n
Â
()
=
(
¢
()
)
=
(
()
)
¢
()
¢
()
QQ
g
0
g
m
0
m
0
m
0
.
(9.37)
I
I
ij
i
j
ij
,
=
1
Definition.
The metric coefficients g
ij
are also called the
coefficients of the first
fundamental form
.
Clearly, the matrix (g
ij
) of metric coefficients with respect to a parameterization
F is just the matrix of the symmetric bilinear map associated to the quadratic form
Q
I
with respect to the basis consisting of the vectors and g is the discriminant
of Q
I
with respect to that basis. Because we have a positive definite form, it follows
that g > 0 (Corollary 1.9.13).
In the case of a surface
S
the coefficients of the first fundamental form have his-
torically been given the names E, F, and G, that is,
∂∂
F
u
i
(
)
=
(
)
(
)
=
(
)
(
)
=
(
)
Euvguv Fuvguv dGuvguv
,
,
,
,
,
,
,
,
,
(9.38)
11
12
22
so that, letting m(t) = (u(t),v(t)), equation (9.37) is often written as
I
g
()
=¢
2
2
Q
u
+ ¢
2
F u v
¢ +¢
v
.
(9.39)