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)
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