Graphics Reference
In-Depth Information
Equations (9.53) complete our analysis of D
n
and we are ready to draw some
important consequences. Incidentally, equations (9.48) with the values of c
ij
as shown
in (9.53) are called the
Weingarten equations
.
9.9.15. Theorem.
The Gauss curvature K satisfies
2
2
LN
-
-
M
LN
-
¥
M
K
=
=
.
2
2
FF
EG
F
u
v
Proof.
The theorem follows from equation (9.52) and the definition of Gauss
curvature as the determinant of matrix (9.50). We see that K is just the quotient
of the discriminants of the first and second fundamental forms.
9.9.16. Theorem.
The mean curvature H satisfies
EN
+-
-
GL
2
FM
H
=
.
(
)
2
2
EG
F
Proof.
The theorem follows from the definition of H and the values in (9.53) from
which we can immediately compute the trace of the matrix (9.50) for D
n
.
Finally, we relate the Gauss and mean curvatures to the principal normal curva-
tures k
1
and k
2
. Since the -k
1
and -k
2
are eigenvalues of D
n
, the linear map D
n
+ kI
is not invertible when k = k
1
or k
2
. Therefore, its matrix has zero determinant, that is,
ckc
c
+
Ê
Ë
ˆ
¯
11
12
det
=
0
.
c
+
k
21
22
This expands to
2
(
)
(
)
=
k
++
c
c k cc
+
-
cc
0
,
11
22
11
22
12
21
or
2
k
-
2
kK
+
=
0
,
(9.54)
for k = k
1
or k
2
. In other words, we have proved
9.9.17. Theorem.
The principal normal curvatures satisfy Equation (9.54) and are
given by the formula
2
HHK
±
-
.
(9.55)
It is again time for an example. Analyzing the geometry invariably involves com-
puting the functions that are the coefficients of the first and second fundamental
forms. The functions L, M, and N are the more complicated. They were defined in the
derivation of equation (9.47). Each had basically two definitions. However, given the