Graphics Reference
In-Depth Information
where G
ij
and a
ij
are some constants. One can show (by taking the dot product of
equations (9.28) with F
u
and F
v
) that these constants are given by the following
formulas:
a
=
L
,
a
=
a
=
M
,
a
=
N
,
11
12
21
22
GE
-
2
FF
+
FE
GE
-
-
FG
2
GF
-
GG
-
FG
u
u
v
v
u
v
u
v
1
1
1
1
2
1
2
1
G
=
,
G
==
G
,
G
=
,
(
)
(
)
(
)
2
2
2
2
EG
-
F
2
EG
F
2
EG
-
F
2
EF
-
EE
+
FE
EG
-
FE
EG
-
2
FF
+
FG
u
v
u
u
v
v
v
u
1
2
1
2
2
2
2
2
(9.59)
G
=
,
G
==
G
,
G
=
.
(
)
(
)
(
)
2
2
2
2
EG
-
F
2
EEG
-
F
2
EG
-
F
Definition.
The quantities G
ij
are called the
Christoffel symbols
of the surface
S
for
the parameterization F(u,v).
Not only do we have the formulas above for the Christoffel symbols, but since
they are functions of E, F, and G and their derivatives, it follows that
any quantity
defined in terms of these symbols will be invariant under isometries
!
We get some additional constraints on the Christoffel symbols if we use the fact
that C
3
functions have mixed partials that can be obtained by differentiating in any
order. In particular,
()
=
()
()
=
()
.
F
F
and
F
F
(9.60)
u
uv
u
v
v
vu
uv
vu
One can show that equations (9.60) hold if and only if
(
)
-
1
1
1
2
1
1
1
2
LML
-=
G
+
M
G
-
G
N
G
v
u
(
)
-
2
1
2
2
1
1
1
2
(9.61)
ML L
-=
G
+
M
G
-
G
N
G
v
u
and
-=
(
[
)
-
()
+
]
2
2
2
1
2
1
2
LN
M
F
G
G
G
G
-
G
G
22
12
22
11
12
12
u
v
[
]
(
)
-
()
+
2
1
1
1
2
1
1
1
2
2
1
1
1
1
1
1
1
2
2
1
(9.62)
+
E
G
G
GG
+
GG
-
GG
-
GG .
u
v
Equations (9.61) are called the
Mainardi-Codazzi equations
. Equation (9.62) is called
the
Gauss equation
. Equations (9.61) and (9.62) together are called the
compatibility
equations
.
The Gauss equation has an important consequence. Since the Christoffel symbols
G
ij
depend only on E, F, G, and their derivatives, the quantity LN - M
2
depends only
on the coefficients of the first fundamental form and their derivatives. It follows from
Theorem 9.9.15 that the Gauss curvature depends only on the coefficients of the first
fundamental form (even though it was defined from the second fundamental form).
Therefore we have proved one of the most fundamental results in the theory of sur-
faces, namely,
9.9.27. Theorem.
(Theorema Egregrium) The Gauss curvature K(
p
) of a surface of
class C
2
is an isometric invariant.
